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A posteriori error analysis of stabilised FEM for degenerate convex minimisation problems under weak regularity assumptions

A posteriori error analysis of stabilised FEM for degenerate convex minimisation problems under... cc@math.hu-berlin.de Department of Mathematics, Background: The discretisation of degenerate convex minimisation problems Humboldt-Universität zu Berlin, experiences numerical difficulties with a singular or nearly singular Hessian matrix. Unter den Linden 6, 10099, Berlin, Germany Methods: Some discrete analog of the surface energy in microstrucures is added to Department of Computational the energy functional to define a stabilisation technique. Science and Engineering, Yonsei University, Unter den Linden 6, Results: This paper proves (a) strong convergence of the stress even without any 120-749, Seoul, Korea smoothness assumption for a class of stabilised degenerate convex minimisation problems. Given the limitted a priori error control in those cases, the sharp a posteriori error control is of even higher relevance. This paper derives (b) guaranteed a posteriori error control via some equilibration technique which does not rely on the strict Galerkin orthogonality of the unperturbed problem. In the presence of L control in the original minimisation problem, some realistic model scenario with piecewise smooth exact solution allows for strong convergence of the gradients plus refined a posteriori error estimates. This paper presents (c) an improved a posteriori error control in this interface problem and so narrows the efficiency reliability gap. Conclusions: Numerical experiments illustrate the theoretical convergence rates for uniform and adaptive mesh-refinements and the improved a posteriori error control for four benchmark examples in the computational microstructures. Keywords: Adaptive finite element method; Relaxation; Convexification; Calculus of variations; Degenerate convex problems; Energy reduction; Nonconvex minimisation; Partial differential equation; Stabilisation; Strong convergence; A posteriori error estimate; Reliability-efficiency gap; Euler-Lagrange equation; Guaranteed upper bound Background Infimising sequences of variational problems with non-quasiconvex energy densities, in general, develop finer and finer oscillations with no classical limit in Sobolev func- tion spaces called microstructure [1-6]. Those oscillations cause difficulty to numerical methods because fine grids are necessary to resolve such oscillations which results in ineffective and tricky mesh-depending computations. Strong convergence of gradients of infimising sequences of the non-quasiconvex problem is impossible. Relaxation techniques replace the nonconvex energy density by its (semi-)convex hull and lead to a macroscopic model. Since the convexified energy density obtained by this © 2013 Boiger and Carstensen; licensee Springer. This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. Boiger and Carstensen Advanced Modeling and Simulation in Engineering Sciences 2013, 1:5 Page 2 of 23 http://www.amses-journal.com/content/1/1/5 method, in general, lacks strict convexity, numerical algorithms might encounter sit- uations where the Hessian matrix is singular. For instance, the Newton minimisation algorithm fails on the convexified three-well problem of Subsection ‘Three-well bench- mark’ below. Applications of relaxation techniques include models in computational microstructure [5-7], some optimal design problems [8,9], the nonlinear Laplacian [10] (where the Hessian can become arbitrarily ill-conditioned in spite of its strict convexity) and elastoplasticity [1]. Stabilisation techniques regularise the energy term by an additional positive semidefi- nite stabilisation function. The paper [11] discusses several choices of such stabilisation functions for P conforming finite elements and quasiuniform meshes. It turns out that stabilisation can ensure strong convergence of the strain approximations under particu- lar circumstances. A particular stabilisation in [12] leads to strong convergence even on unstructured grids but is still restricted to unrealistically smooth solutions. This paper studies the stabilisation technique of [12] and addresses the question of convergence (i) without extra regularity assumptions, (ii) in a realistic scenario called model interface problem, and (iii) establishes an a posteriori error control. The stabilisation leads to improved condition numbers of the Hessian matrix and to reduced errors if the numerical solvers fail without stabilisation. Figure 1 shows the con- vergence of the discrete stress σ of the three-well benchmark corresponding to the discrete minimisers of the energy E (v ) = E(v ) + C/2|v | . The errors are plot- ted for computations with uniform mesh refinements with various solver tolerances in the discrete minimisation procedure at a fixed triangulation and values of C,cf. Section ‘Numerical experiments’ for details on the MATLAB implementation. Without −5 stabilisation, the convergence stagnates with a moderate tolerance of 10 which becomes visible as a “plateau” in Figure 1. The Newton solver even aborts prematurely due to the singular Hessian. In conclusion, stabilisation enables higher accuracies in numerical examples. For β  0 the convex energy functional assumes the form E(v) := W (Dv(x)) + β|v(x) − g(x)| − f (x) · v(x) dx. (1.1) Assume that W is convex with quadratic growth so that there exist minimisers u ∈ H ();below p-th order growth is included while p = 2 throughout this simplifying Figure 1 Impact of the stabilisation on the error. Error of the stabilised stress σ with coefficients −4 −5 −6 −8 C = 0, 10 , 1 of the stabilisation and various tolerances tol = 10 ,10 ,10 of the Newton solver. Boiger and Carstensen Advanced Modeling and Simulation in Engineering Sciences 2013, 1:5 Page 3 of 23 http://www.amses-journal.com/content/1/1/5 introduction. Given a sequence of shape-regular triangulations (T ) [13], let u ∈N minimise the stabilised discrete energy 2 2 2 −1 2 E (v ) := E(v ) + |v | with |v | := H h || [Dv ] || F 2 L (F) F∈F () amongst all conforming P finite element functions v on T ,where [Dv ] is the jump of 1    F the gradient Dv along the interior side F,written F ∈ F (),and H := max h is the T T maximal diameter h of all simplices T ∈ T . Section ‘Global convergence’ verifies the strong convergence of the discrete solution u and its stress σ := DW (Du ) to their respective continuous conterparts, 2 2 2 ||σ − σ || + β||u − u || +|u | → 0as  →∞. 2  2 L () L () Section ‘A posteriori error estimates’ presents a novel application of [14-17] to non- linear problems. For the L projection  onto the space of piecewise P functions, any Raviart-Thomas function τ ∈ RT (T ) satisfies ||σ − σ || L () ||σ − τ || 2 +||  + div τ || 2 + osc ( ) ||u − u || 1 . ,2 L () L () H () This error bound holds for any discrete displacement u that satisfies the boundary con- ditions; the point is that inexact solve is included — there is no Galerkin orthogonality required. The drawback is to minimise the expression on the right-hand side with respect to τ in order to obtain a sharp error bound. This is a particular selection: degenerate con- vex minimisation problems do not allow for a control of ||u − u || 1 and may even face H () multiple exact or discrete solutions while the discrete minimum of E is unique. However, in some results of this paper, either W or the lower-order terms lead to some control over ||u − u || 2 and the selection via stabilisation is correct. H () Phase transition problems motivate the investigation of scenarios with a smooth solu- tion u up to a one-dimensional interface ⊂  [18]. Section ‘Refined analysis for an interface model problem’ proves that such problems allow even for strong convergence of 1,∞ 2 the gradients for any unique solution u in W ()∩H (\ ) [19]. This result also leads to an improvement of the a posteriori error control of the discrete stresses and narrows the efficiency-reliability gap; the efficiency-reliability gap is the difference of the conver- gence rates of the guaranteed upper a posteriori error bound and the guaranteed lower a posteriori error bound. Section ‘Numerical experiments’ complements the theoretical findings with numerical experiments to provide empirical evidence of the improved error control. The sta- bilisation technique competes in four benchmark examples, with and without known exact solution, for uniform and two different mesh-refining algorithms for the explicit residual-based error estimator of [7] and with an averaging-type error estimator of ([18], (1.11)). Standard notation on Lebesgue and Sobolev spaces is employed throughout this paper and a  b abbreviates a  Cb with some generic constant 0 < C < ∞ independent of crucial parameters (like the mesh-size on level ); a ≈ b means a  b  a.Furthermore, A : B abbreviates the matrix inner product that corresponds to the Frobenius norm. Boiger and Carstensen Advanced Modeling and Simulation in Engineering Sciences 2013, 1:5 Page 4 of 23 http://www.amses-journal.com/content/1/1/5 Methods: Discretisation and Stabilisation Based on the convergence results for unstructured grids, this paper will develop reliable error estimators for a class of stabilised convex minimisation problems described in the sequel. Let  ⊂ R be a bounded Lipshitz domain with polygonal boundary for n = 2or m×n 2 m 3. Given a continuous convex energy density W : R → R, g, f ∈ L (; R ), β  0, 1,p m and v ∈ W (; R ) with 2  p < ∞ and m = 1, ... , n, the energy is given by (1.1). 1 m×n Throughout this paper, the energy density W ∈ C (R ; R) satisfies (2.1)–(2.2) for parameters 1 < r  2, 0  s < ∞ and s + r + p  rp.The two-sided growth condition reads p p m×n |F| − 1  W (F)  |F| + 1 for all F ∈ R . (2.1) m×n The convexity control assumption reads, for all F , F ∈ R , 1 2 r s s |D W (F ) − D W (F )|  1 +|F | +|F | (D W (F ) − D W (F )) : (F − F ). 1 2 1 2 1 2 1 2 (2.2) The proof of Theorem 2 in [7] shows that (2.2) is crucial for the uniqueness of the stress tensor DW (Du). 2,p m 2 m Given Dirichlet data u ∈ W (; R ) ∩ H (∂; R ) for the set of admissible func- 1,p tions A := u + V := u + W (; R ), the continuous (convex) model problem D D reads minimise E(v) within v ∈ A. (2.3) A finite element approximation of (2.3) is based on a family of regular triangulations (T ) of the domain  into simplices in the sense of Ciarlet [13] (e.g., for n = 2, two ∈N non-disjoint triangles of T shareeitheracommonedgeoracommonnode).The setof sides F consists of edges (for n = 2) or faces (for n = 3) of T and is split into the union of the sets of all interiour sides F () and of all boundary sides F (∂). For latter reference, define the diameter h := diamT of a triangle (or tetrahedron) T ∈ T and the size h := diamF of a side F ∈ F .The mesh size function h :  → R F   >0 is given by h for x ∈ int T ∈ T , h (x) := min {h : F ∈ F and x ∈ F} otherwise. Theglobalmeshsizewillbeabbreviatedby H :=h  ∞ .Wepresume thefamily L () (T ) to be shape-regular so that h ≈ h for all T ∈ T , F ∈ F and F ⊂ T. ∈N F T The space of T -piecewise polynomials of degree  k ∈ N is P (T ).The nodal 0 k interpolation I w ∈ P (T ) ∩ C() of w ∈ C() is given by I w(z) = w(z) for all 2 2 nodes z.Let furthermore  w be the L projection of w ∈ L () onto P (T ),and osc (w) :=h (id −  )w be the oscillation of w ∈ L () for 2  q  ∞ ,q   L () with respect to the triangulation T . The symbol id denotes the identity operator. Let u = I u ,and D,  D A := u + V with V := V ∩ P (T ; R ) ∩ C(). D,   1 Given a function v on  which is possibly discontinuous along some side F ∈ F () shared by the two elements T such that there exist traces from either sides, the jump of v along F reads Boiger and Carstensen Advanced Modeling and Simulation in Engineering Sciences 2013, 1:5 Page 5 of 23 http://www.amses-journal.com/content/1/1/5 [v] (x) =[v] (x) := lim v( y) − lim v( y) for x ∈ F. T y→x T y→x + − The stabilisation of [12] will be used throughout this paper with −1 <γ < ∞ and 1+γ a (v, w) := [Dv] :[Dw] ds and |v| := a (v, v). (2.4) F F F F F∈F () The stabilised discrete problem reads minimise E (v) := E(v) + a (v, v) amongst v ∈ A . (2.5) Convergence of gradients with a guaranteed convergence rate is shown in [12] under unrealistically high regularity assumptions. A comprehensive collection of the results in [12] is summarised in the following theorem. 3/2+ε m Theorem 2.1. ([12])Let u ∈ A ∩ H (; R ) be some solution of (2.3) for some ε> 0; let p and r be the Hölder conjugate of p and r, −1 <γ < 3, and set ζ := min 1 + γ , r for β> 0 and ζ := min {1 + γ ,2} for β = 0. Then the discrete solution u ∈ A of (2.5) and the continuous and discrete stress σ := p m×n m×n (; R ) and σ := DW (Du ) ∈ P (T ; R ) satisfy DW (Du) ∈ L (1+γ)/2 ζ r 2 2 2 σ − σ  +u − u  + |||u ||| + H D(u − u )  H . 2   2 L ()  L () L () Proof. This combines Lemma 3.5 and 4.1–4.2 plus Theorem 3.8 and 4.4 in [12]. Global convergence This section is devoted to the proof of a general convergence result without higher reg- ularity assumptions. Let u ∈ A and u ∈ A solve the minimisation problem (2.3) and (2.5) and set σ := DW (Du) and σ := DW (Du ). For the unstabilised approximation, the a priori error estimates of [7] plus a density argument prove convergence of r 2 σ − σ  + βu − u  → 0as H → 0. L () L () Note that β = 0 is permitted. Then, however, uniqueness of u and convergence of u − u  are guaranteed. The point in the following result is that the stabilised approxi- L () mation converges as well as |||u ||| → 0 even for non-smooth or non-unique minimisers. Under special circumstances, uniqueness of u and the convergence u − u  2 → 0 L () can be shown even for β = 0, e.g., in Example 3.3. Theorem 3.1. (Global Convergence)Provided lim H = 0 it holds →∞ r 2 2 σ − σ  + βu − u  + |||u ||| → 0 as  →∞. L () L () The proof is based on the following lemma. Lemma 3.2. The errors δ := σ − σ and e := u − u satisfy, for all v ∈ V ,that r 2 r 2 δ  + βe   |e − v | + βe − v | + a (u , v ). 2   1,p   2 L () W () L () L () Proof. The minimisation problems (2.3) and (2.5) are equivalent to their respective Euler-Lagrange equations, namely for v ∈ V and v ∈ V , σ(x) :Dv(x) + 2β(u(x) − g(x)) · v(x) − f (x) · v(x) dx = 0; (3.1) Boiger and Carstensen Advanced Modeling and Simulation in Engineering Sciences 2013, 1:5 Page 6 of 23 http://www.amses-journal.com/content/1/1/5 σ (x) :Dv (x) + 2β(u (x) − g(x)) · v (x) − f (x) · v (x) dx + a (u , v ) = 0. (3.2) Algebraic transformations of the difference of these two equations lead to ˆ ˆ δ :De dx + 2βe  = (δ :D(e − v ) + 2βe · (e − v ))dx + a (u , v ). L () It is shown in ([12], Lemma 3.5) that δ   δ :De dx. (3.3) L () Two Hölder inequalities on the right-hand side and absorbtions of δ and L () e  2 eventually concludethe proof. Furtherdetails aredropped forbrevity. L () Proof of Theorem 3.1. Given any positive ε, the density of smooth functions in 1,p m m W (; R ) leads to some v ∈ D(; R ) such that u − u − v   ε.Hence 1,p ε D ε 0 W () v := I (v + u ) − u ∈ V satisfies ε D e − v = (u − u − v ) + (id − I )(v + u ). D ε  ε D Note that the nodal interpolation I (v + u ) is well-defined since v and u are assumed ε D ε D to be smooth. With ([12], Lemma 3.1–3.2) it follows that (id − I )(v + u ) 1,p  H → 0and ε D W () 1+γ 2 2 |||I (v + u )||| = |||(id − I )(v + u )|||  H → 0as  →∞. ε D  ε D Since · 2  · 1,p , this yields some  ∈ N such that L () W () r 2 2 |e − v | + βe − v  () + |||I (v + u )|  ε for all    . 1,p   2  ε D 0 W () L A Cauchy inequality applied to the stabilisation norm proves 1 1 2 2 2 a (u , v ) = −|||u ||| + a (u , I (v + u ))  − |u | + |I (v + u )| . ε D   ε D 2 2 Substitute a (u , v ) in Lemma 3.2 and add |||u ||| on both sides. This leads to r 2 2 δ  + βe  + |||u |||  ε for all    . 2  0 L () L () Example 3.3. The two-well example from the computational benchmark [18] allows an estimate on e  2 even for β = 0. Let n = 2, let F :=−F := (3, 2)/ 13, and let the L () 1 2 2 2 energy density W be theconvexhull of F →|F − F | |F − F | .Thatis 1 2 2 2 2 W (F) = max 0, |F| − 1 + 4 |F| − (3F(1) + 2F(2)) /13 . (3.4) Then ([11], Lemma 9.1) proves, for all v ∈ V ,that 2 2 e   δ :De dx +e − v  . 2     1 L () H () Therefore, the arguments of Lemma 3.2 lead to r 2 r 2 δ  +e   |e − v | +e − v  + a (u , v ). 2   1,p   1 L () W () H () L () This result can be used in the proof of Theorem 3.1 in order to obtain r 2 2 σ − σ  +u − u  + |||u ||| → 0as  →∞. L () L () Boiger and Carstensen Advanced Modeling and Simulation in Engineering Sciences 2013, 1:5 Page 7 of 23 http://www.amses-journal.com/content/1/1/5 A posteriori error estimates Beyond the a posteriori error analysis of [7], the additional stabilisation term in the dis- cretisation of this paper causes an additional difficulty in that the Galerkin orthogonality does not hold for the natural residual. Inspired from novell developments in the a posteri- ori error control of elliptic PDEs motivated by inexact solve [14-17], this section presents some guaranteed upper error bound for the discretisation at hand for any approximation u which does not necessarily satisfy (3.2) exactly. Thereby inexact solve is included. Let u ∈ A solve (2.3) and let u ∈ A be arbitrary. It is not assumed that u solves the discrete problem (2.5); the following theorem holds regardless of this. Recall the def- initions of osc (·) and  from Section ‘Methods: Discretisation and Stabilisation’ and ,q given σ := DW (Du) and σ := DW (Du ), abbreviate :=−2β(u − g) + f , e := u − u and δ := σ − σ . 1,p m Theorem 4.1. Given any w ∈ W (; R ) with w = u − u on the boundary ∂,and m×n given any τ ∈ H(div, ; R ), it holds, for all 2  q  p and for some constant κ known from ([12], Lemma 3.5), that r 2 1−r r 2 κ/2δ  + βe   (rκ/2) /r |w | + βw 2  1,p  2 L () W () L () L () + σ − τ  +  + divτ  + osc ( ) e − w  1,q . q   q ,q W () L () L () The constant κ depends on problem-specific data such as u 1,p and the size of the W () domain . Refer to the proof of Lemma 3.5 in [12] for details. Before the proofs conclude this section, some practical choice of τ in Theorem 4.1 is discussed as some Raviart-Thomas finite element functions in RT (T ) := τ ∈ P (T )∩H(div, ) : ∀T ∈ T ∃a, b, c ∈ R ∀x ∈ T, τ (x) = (a, b)+cx . 0  RT 1   RT We suggest the computation (or an accurate approximation) of μ := min σ − τ +  + divτ (4.1) q q L () L () τ ∈RT (T ) and emphasise that any upper bound is allowed in Theorem 4.1. This leads to r 2 1−r r 2 κ/2δ  + βe   (rκ/2) /r |w | + βw 2 1,p 2 L () W () L () L () + μ + osc ( ) e − w  1,q . ,q W () The algorithm of ([20], Prop. 4.1) computes some w from (id − I )u with 1/q q q w  ≈ h (id − I )u  and (4.2) L (T ) T  D L (∂T ∩∂) 1/q−1 1/q q q q Dw   h (id − I )u  + h ∂(id − I )u /∂s . L (T )  D L (∂T ∩∂)  D L (∂T ∩∂) T T (The proof of the second assertion is analogous to that of ([20], Prop. 4.1) and the first is an immediate consequence of the design of w ). This and e − w  1,q  1for W () bounded u (i.e. solely u  1,p  1 is assumed) lead to the practical estimate μ as a W () computable guaranteed upper bound of the left-hand side of Theorem 4.1. Since the min- imisation of (4.1) is computationally intensive for q = 2, Section ‘Numerical experiments’ actually computes an approximation of μ ,based on q = 2. The choice τ = σ in Theorem 4.1 shows that the right-hand side is in fact optimal up to oscillations.The reliability-efficiency gapof[18] is visiblehereinthatwehavenofurther estimate on u  1,p [7,18]. However, additional smoothness assumptions on u may W () lead to refined estimates on the term e − w  1,q (cf. Section ‘Refined analysis for W () Boiger and Carstensen Advanced Modeling and Simulation in Engineering Sciences 2013, 1:5 Page 8 of 23 http://www.amses-journal.com/content/1/1/5 an interface model problem’). The following result indicates that μ is sharp in the sense that it converges with the correct convergence rate. This theorem employs the Fortin t n interpolation operator I defined for τ ∈ H(div, ) ∩ L (; R ) with t > 2by I τ ∈ F, F, RT (T ) and n · (id − I )τds = 0 for all F ∈ F . F F, Here and in the following, n denotes a unit normal vector of the side F; the direction of n arbitrary, but fixed for a given side F. For the improved regularity of stress in the class of degenerate convex minimisation problems at hand, we refer to [3,21]. Theorem 4.2. (Efficiency)Ifthe exactstress σ is sufficiently regular such that its Fortin m×n interpolant τ = I σ ∈ RT (T ; R ) is defined, it holds F, 0 σ − τ +  + divτ q    q L () L () δ  + 2βe  +(id − I )σ  . q  q F, q L  L () L () It is expected that (id − I )σ   H . This is shown in ([22], Prop. 3.6) for F, q L () q = 2 and therefore also holds for q  2. Hence the right-hand side of the assertion of Theorem 4.2 converges with the (expected) optimal convergence rates. Proof of Theorem 4.1. Let κ be the reciprocal of c in ([12], Lemma 3.5), which is also the multiplicative constant hidden in (3.3). Recall Young’s inequality, which reads ab r r a /r + b /r for a, b > 0. This, (3.3) and the continuous Euler-Lagrange equation (3.1) show, for v = e − w ∈ V,that r 2 κδ  + 2βe   (δ :Dv + 2βe · v)dx L () L () + (δ :Dw + 2βe · w )dx − (σ :Dv −  · v)dx 2 2 + βe  + βw 2  2 L () L () r 1−r + κ/2δ  + rκ/2 /r |w | . ( ) 1,p W () L () Hence Res (v) :=− (σ :Dv −  · v)dx satisfies r 2 1−r r 2 κ/2δ  + βe   Res (v) + (rκ/2) /r |w | + βw  . 2 1,p 2 L () W () L () L () 1,q Let C denote the Poincaré constant of convex domains with respect to the W norm. The fundamental theorem of calculus on some one-dimensional arc shows that C  1. The paper [23] proves C = 1/2. Hence, operator-interpolation arguments [24,25] prove 1/q C  (1/2)  1. The Poincaré inequality shows, for any 2  q  p,that ˆ ˆ (id −  ) · vdx = h (id −  ) · (id −  )vdx h (id −  ) Dv q = osc ( )Dv q . q L () ,q  L () L () m×n For any τ ∈ H(div, ; R ), the Hölder and Poincaré inequalities show Res (v) =− (σ − τ) :Dv − (  + divτ) · v − (id −  ) · v dx σ − τ  +  + divτ  + osc ( ) v 1,q . q   q ,q W () L () L () Boiger and Carstensen Advanced Modeling and Simulation in Engineering Sciences 2013, 1:5 Page 9 of 23 http://www.amses-journal.com/content/1/1/5 Proof of Theorem 4.2. The triangle inequality yields σ − τ (id − I )σ +δ q q q F, L () L () L Since f = 2β(u − g) − divσ , the commutative property divI =  div (cf. ([22], p. 129)) F, yields + divτ  = 2β e   2βe  . q   q  q L () L () L () Refined analysis for an interface model problem This section is devoted for a model scenario from phase transition problems [18] with some solution u that is smooth outside some one-dimensional interface .Sup- pose some (possibly non-unique) minimiser u of the continuous problem (2.3) satisfies 1,∞ m 2,p m u ∈ W (; R ) ∩ W ( \ ; R ) for some finite union of (n − 1) dimensional Lipschitz surfaces in .Since  has a Lipschitz boundary, this implies Lipschitz continu- 1,∞ m ity of u on . We refer to [19] for sufficient conditions for u ∈ W (; R ) and conclude 2,p m that the remaining assumption u ∈ W ( \ ; R ) is the essential hypothesis expected in many interface problems. Let u ∈ A be the(unique)minimiser of thediscretesta- bilised problem (2.5). In the following, also =∅ is permittedtoextendpreviousresults [12] for highly regular minimisers. The following theorem leads to a priori convergence rates for the interface model prob- lem. Thereby it recovers the results of [12] for problems with piecewise smooth exact solution. We will abbreviate the set of all triangles that are touched by as T ( ) :={T ∈ T : dist(T, ) = 0}, its cardinality as |T ( )|,its unionas  := int( T ( )) with volume | | and its complement as  :=  \  . , , Theorem 5.1. Provided β> 0,itholds 1+γ r/(r−1) r/(r−1) r 2 2 2 2 2 δ  +e  +|u |  H |u| + H |u| +H |u| 2  2 1,∞ p   2,p C L ()  H (\ ) W () L () W ( ) γ +n−1 r/(r−1) 2 r/((r−1)p) + H |u| |T ( )|+|u| | | . 1,∞ 1,∞ W () W () 1−n Remark 5.2. In the case of uniform mesh refinements we may expect |T ( )|≈ H and | |≈ H and Theorem 5.1 simplifies to r/((r−1)p) min {γ ,2} r/(r−1) r 2 2 2 δ  +e  +|u |  H |u| + H |u| . 2  1,∞ p 1,∞ L () W () W () L () Proof. With w = (id − I )e = (id − I )u, a Young inequality, (3.3) and ([12], Theorem 3.8) yield r/(r−1) r 2 2 2 2 δ  +e  +|u |  |w | +w  +|I u| . 2   1,p  2 L () W () L () L () Theorem 4.4.4 in [25] shows w  2  w  ∞  H |u| 1,∞ and L () L () W () p p p |w | =|w | +|w | 1,p  1,p 1,p C W () W ( ) , W ( ) p p p |u| | |+ H |u| . 1,∞ C 2,p W ( ) , W ( ) , Boiger and Carstensen Advanced Modeling and Simulation in Engineering Sciences 2013, 1:5 Page 10 of 23 http://www.amses-journal.com/content/1/1/5 Let ω = T be the patch of a side F ∈ F ,and set F ( )= {F ∈ F () : ω ∩ = F    F T ∈T F⊂T C C ∅} and F ( ) = F () \ F ( ).Notethat[Du] =0for F ∈ F ( ).Then ⎛ ⎞ 1+γ ⎜ ⎟ 2 −1 2 −1 2 |I u| = H h [Dw ]  + h [DI u]  · ⎝ ⎠ F 2  F 2 F L (F) F L (F) F∈F ( ) F∈F ( ) The first sum can be estimated as in the proof of ([12], Lemma 3.2), the second sum with 2 n−1 2 n−1 2 [DI u]   h |I u|  h |u| . F 2  1,∞ 1,∞ L (F) F W (F) F W (F) The observation |F ( )|  (n + 1)|T ( )| concludes the proof. Together with Theorem 5.1, the subsequent result implies strong convergence of the gradients in the model interface problem as H → 0. Theorem 5.3. Under the aforementioned conditions on the (possibly non-unique) exact 1,∞ m 2,p m minimiser u ∈ W (; R ) ∩ W ( \ ; R ),the errore = u − u of the discrete solution u ∈ A of (2.5) satisfies 1/3 5/6 1/3 (1−γ)/2 2 2 De  2 e  + H ∂ u /∂s  + H |u | L ()  2 D 2 L () L (∂) −(1+γ)/4 1/2 1/2 5/4 1/2 2 2 + H |u | e  + H ∂ u /∂s  . 2 2 L () L () Proof. The basic idea of gradient control is the generalisation of the interpolation esti- 1 2 2 mate H () = [L (), H ()] for a reduced domain \ ; refer to [24,25] for a detailed 1/2 analysis of interpolation spaces. Let w be the boundary value interpolation of (id − I )u as described in ([20], Prop. 4.1), such that w satisfies the inequalities in (4.2). A piecewise 1,p integration by parts shows, for v := e − w ∈ W (; R ),that ˆ ˆ De  = D(u − u ) :Dvdx + De :Dw dx L () ˆ ˆ ˆ v · [Du] n ds − v · u dx − v · [Du ] n ds F F \ F F∈F () +De  2 Dw  2 , L () L () where n is a unit normal vector of the interface . The Lipschitz continuity of u implies |[Du] n |  1. This and the trace inequality on lead to 1/2 1/2 v · [Du] n ds  v 2  v 2 +v Dv . L ( ) L () 2 2 L () L () The case = ∅ is contained in ([12], Theorem 4.4). The piecewise Laplacian of u is bounded in L () and so (with the generic constant C :=u 2 hidden in the L (\ ) notation C ≈ 1) v · udx  v 2 L () The elementwise trace inequality ([25], Theorem 1.6.6, p. 39) for an n-dimensional 1,q m simplex T and one of its sides F,and f ∈ W (T; R ),1  q < ∞,reads q q q−1 q q−1 q −1 −1 f   h f  +f  Df  q  h f  + h Df  . q q q q q L (T ) L (F) T L (T ) L (T ) T L (T ) T L (T ) Boiger and Carstensen Advanced Modeling and Simulation in Engineering Sciences 2013, 1:5 Page 11 of 23 http://www.amses-journal.com/content/1/1/5 The term v · [Du ] n ds and the stabilisation |u | are already analysed in the Estimate on C in the proof of ([12], Theorem 4.4). This results in (1−γ)/2 −(1+γ)/2 v · [Du ] n ds  |u | H Dv 2 + H v 2 . F L () L () F∈F () The preceding estimates plus the absorbtion of De  2 lead to L () 1/2 1/2 2 2 De   v +v Dv +Dw 2 2 2  2 L () L () L () L () L () (1−γ)/2 −(1+γ)/2 +|u | H Dv 2 + H v 2 . L () L () The triangle inequality applied to v = e − w and some careful elementary analysis to 1/2 absorb De  eventually lead to L () 1/3 1/3 (1−γ)/2 De  2  e  +w  +|w | 1 + H |u | L ()  2  2  H () L () L () 1/2 −(1+γ)/4 1/2 + H |u | e  2 +w  2 . L () L () The inequalities (4.2), Poincaré and Friedrichs inequalities on sides F ∈ F (∂) and removal of higher-order terms in H conclude the proof. The following theorem is an improved a posteriori estimate based on Theorems 4.1 and 5.3. 1,∞ m 2,p m Theorem 5.4. Recall u ∈ W (; R ) ∩ W ( \ ; R ), the definitions e := u − u and δ := σ − σ for σ := DW (Du) and σ := DW (Du ), and the definition of  from Section ‘A posteriori error estimates’. Set M(τ ) :=σ − τ  2 +  + divτ  2 + osc ( ) ,2 L () L () m×n for all τ ∈ H(div, ; R ) . Provided β> 0,itholds −(1+γ)/3 2/3 r 2 6/5 4/3 δ  +e   M(τ ) + H M(τ ) |u | L () L () (1−γ)/2 1−γ/4 1/2 min {5,r (1+1/p)} + M(τ ) H |u | +H |u | +H and min {5/3,r (1+1/p)/3} −(1+γ)/9 2/9 2 2/5 4/9 De   M(τ ) + H M(τ ) |u | + H L () 1/3 (1−γ)/2 1−γ/4 1/2 1−γ 1/3 2 + M(τ ) H |u | + H |u | + H |u | 1/2 min 5,r (1+1/p) −(1+γ)/2 −(1+γ)/3 2/3 { } 6/5 4/3 + H |u | M(τ ) + H M(τ ) |u | + H 1/2 −(1+γ)/2 (1−γ)/2 1−γ/4 1/2 1/2 + H |u | M(τ ) H |u | + H |u | The generic constants in Theorem 5.4 depend on problem-specific data such as the shapes of  and as well as the generic constant κ of Theorem 4.1. Theorem 5.5. Theorem 5.4 holds verbatim in Example 3.3 and in the modified two-well problem of Subsection ‘Modified two-well benchmark’, where β = 0. Remark 5.6. The assertion of Theorem 5.4 holds for any discrete u ∈ u + V which D, may approximate the discrete unique exact solution of (2.5). This allows the inexact SOLVE via an iterative procedure. Boiger and Carstensen Advanced Modeling and Simulation in Engineering Sciences 2013, 1:5 Page 12 of 23 http://www.amses-journal.com/content/1/1/5 Proof of Theorem 5.4. Choose w as in the proof of Theorem 5.3. Then Theorem 4.1 with q = 2 and (4.2) imply r 2 r 2 δ  +e   M(τ )e − w  1 +|w | +w 2    1,p  2 p H () L () W () L () L () min 5,r (1+1/p) 3/2 { } M(τ ) |e | +e  + H + H . 1 2 H () L () Theorem 5.3 provides an estimate of the semi-norm |e | 1 . A Young inequality shows H () 5/6 5 6 H M(τ )  H +M(τ ) . The absorbtion of e  2 then proves the first assertion. The L () second assertion is an immediate consequence of the first one, Theorem 5.3 and several algebraic transformations. Numerical experiments This section illustrates the theoretical estimates and their impact on the reliability- efficiency gap on 2D benchmarks in computational microstructures [18,26]. Numerical algorithms The adaptive finite element method (AFEM) and algorithmic details on the implementa- tion in MATLAB in the spirit of [27] concern the state-of-the-art AFEM loop SOLVE → ESTIMATE → MARK → REFINE and are explained below together with some notation. Solve The stabilised discrete problem (2.5) is solved in a nested iteration on a given triangulation T with MATLAB’s standard-minimiser fminunc with default tolerances. Gradient and Hessian of the discrete energy are available and therefore provided to fminunc.Weset γ = 1 in the stabilisation term (2.4) in all our experiments. This is motivated by ([12], Theorem 4.4) which suggest that γ = 1 yields an optimal convergence rate. The discrete solution of the previous AFEM loop iteration serves as a start vector for fminunc;for the first iteration, the initial vector is zero everywhere up to the Dirichlet boundary nodes. Since the Galerkin orthogonality is not required in Theorem 4.1, the termination of an iterative realisation for SOLVE is not a sensitive issue. In the computational PDEs, it is a fundamental issue to involve inexact solve. In this paper, however, the numerical examples are run with the standard settings of MATLAB. Estimate The refinement indicator results from the error estimator of Theorem 4.1. As in the work of Repin [28], the computation of the minimiser τ ∈ RT (T ) of σ − τ  2 +  + divτ  2 (6.1) L () L () runs Algorithm 1 based on the formula 2 2 2 (a + b) = min (1 + s)a + (1 + 1/s)b for a, b > 0 s>0 The stopping criterion of Algorithm 1 monitors relative changes and avoids degenerate values of s. Undisplayed experiments have conviced us that a maxmium of three iterations 0.8 and a stopping tolerance of ε (with the machine precision ε ) yield satisfying results. M Boiger and Carstensen Advanced Modeling and Simulation in Engineering Sciences 2013, 1:5 Page 13 of 23 http://www.amses-journal.com/content/1/1/5 Algorithm 1 Approximate flux computation Input: σ , s = 1; for k = 1, 2, 3 do Compute minimiser τ of 2 2 M(s , τ) = (1 + s )σ − τ  + (1 + 1/s )  + divτ  ; k k 2 k 2 L () L () if D (s , τ ) nearly singular (MATLAB “warning”) then return τ ; k k k s =  + divτ  /σ − τ  ; 2 2 k+1   k  k L () L () |s −s | k+1 k 0.8 if max s ,1/s , <ε then return τ ; k+1 k+1 k s +s M k+1 k Output: approximate flux τ The iteration is stopped whenever s,1/s or the relative change of s drops below this toler- ance. As an additional precaution, the iteration also stops if the linear system is deemed “nearly singular” by MATLAB. Our experiments convinced us that ignoring this warn- ing causes a breakdown with NaNs. Note that if q = 2, we still minimise the L sums in (6.1) to avoid the computational cost of a nonlinear solve. With the computed minimiser τ, Section ‘A posteriori error estimates’ yields the error estimator η :=σ − τ  +  + divτ  + osc ( ) . F,q  q   q ,q L () L () This will be compared with the well-established residual based a posteriori error estimator [7] ⎛ ⎞ ⎛ ⎞ 1/q 1/q q q q ⎝ ⎠ ⎝ ⎠ η := h   + h [σ ] · n  , R,q F  F q T q L (F) L (T ) T ∈T F∈F () which is reliable for the original discretisation without stabilisation. Undisplayed exper- iments computed the averaging error estimator [18], which is founded on the same theoretical background as η and therefore yielded essentially the same convergence R,q rates. The error estimators in Theorem 5.4 read 6/5 −(1+γ)/3 4/3 2/3 η := η + H η |u | L,2 F,2  F,2 (1−γ)/2 1−γ/4 1/2 min {5,r (1+1/p)} + η H |u | + H |u | + H F,2 min 5/3,r (1+1/p)/3 2/5 −(1+γ)/9 4/9 2/9 { } η := η + H η |u | + H H,2 F,2 F,2 1/3 1/3 (1−γ)/2 1−γ/4 1/2 1−γ + η H |u | + H |u | + H |u | F,2 1/2 min 5,r (1+1/p) −(1+γ)/2 6/5 −(1+γ)/3 4/3 2/3 { } + H |u | η + H η |u | + H F,2  F,2 1/2 −(1+γ)/2 (1−γ)/2 1−γ/4 1/2 1/2 + H |u | η H |u | + H |u | . F,2    Boiger and Carstensen Advanced Modeling and Simulation in Engineering Sciences 2013, 1:5 Page 14 of 23 http://www.amses-journal.com/content/1/1/5 MARK For any given T ∈ T with its set of faces F (T ), ∂T = F (T ),and given τ from (6.1), set q q q q η (T ) :=σ − τ  +  + divτ  + h (id −  )  . F q q q L (T ) L (T ) L (T ) q q q q /n 1/n η (T ) :=|T |   +|T | [σ ] ·n  . F F R q q L (T ) L (F) F∈F ()∩F (T ) q q Let η (T ) be one of the refinement indicators η (T ) and η (T ). Some greedy algorithm computes M ⊂ T of (almost) minimal cardinality such that q q η (T )  1/2 η (T ). T ∈M T ∈T Refine This step computes the smallest refinement T of T with M ⊂ T \ T based on +1    +1 the red-green-blue refinement strategy as illustrated in Figure 2. This refinement involves some closure algorithm to avoid hanging nodes. Two-well benchmark The computational microstructure benchmark of ([18], Section 2) considers two wells with W from (3.4) in Example 3.3. The energy is given by (1.1) on the domain  = (0, 1)× (0, 3/2) ⊂ R with g(x) for t  0, 5 3 g(x) :=−3t /128 − t /3and u (x) := ⎩ 3 t /24 + t for t  0 for t := (3(x − 1) + 2x )/ 13; p = q = 4and f ≡ 0. The unique minimiser u of 1 2 1,4 min E(v) with A = u + W () reads u = u ([18], Theorem 2.1) and β = 1 allows v∈A D D for Theorems 5.1–5.4 to hold. An initial triangulation T is given by a criss triangula- tion of (0, 1) × (0, 3/2) with 12 congruent triangles and the two interior nodes (1/2, 1/2) and (1/2, 1). The adaptive algorithm of Subsection ‘Numerical algorithms’ computes a sequence of discrete solutions (u ) and stresses (σ ) ,aswellaserror estimators η and η with and without stabilisation for uniform and adaptive meshes and led to Figure 3 with overall observations of Section ‘Conclusions’. The empirical convergence rates for uniform and R- as well as F-adapted mesh-refining are collected in Table 1. Note that the error estimator η performs better than η . This is evident from the table for uniform L F mesh refinements, but a closer look at Figure 3 reveals that even in the adaptive scenarios, Figure 2 Possible refinements of a triangle T. Boiger and Carstensen Advanced Modeling and Simulation in Engineering Sciences 2013, 1:5 Page 15 of 23 http://www.amses-journal.com/content/1/1/5 Figure 3 Convergence plot of the two-well benchmark. Errors and error estimators of the two-well benchmark of Subsection ‘Two-well benchmark’, plotted against the number of degrees of freedom. η converges slightly faster than η .Thisisinaccordance to thetheoryofSection L F ‘Refined analysis for an interface model problem’ where η is derived from η based on L F additional smoothness assumptions. Modified two-well benchmark This subsection concerns a modification of the previous problem with (3.4) and a linear right-hand side for β = 0and f (x) :=−div(DW (Du (x))) and unique solution u = u D D as before. Note that Example 3.3 applies to this problem, and so the proof of Theorem 3.1 yields r 2 2 σ − σ  +u − u  +|u | → 0as  →∞ L () L () and Theorems 5.1–5.4 hold as well. The algorithms of Subsection ‘Numerical algorithms’ ran with and without stabilisation for uniform and adaptive meshes with the same initial triangulation as in Subsection ‘Two-well benchmark’ and led to Figure 4 with overall observations of Section ‘Conclusions’. The empirical convergence rates for uniform and R- as well as F-adapted mesh-refining are collected in Table 1 for completeness although they are almost identical with those observed in Subsection ‘Two-well benchmark’. Three-well benchmark The energy density W of ([26], Example 5.9.3, p. 72) is the convex hull of min{|F| , |F − 2 2 (1, 0)| , |F − (0, 1)| } with explicit form in ([26], Example 5.6.4, p. 58). Let furthermore 2 2 3 = (0, 1) ⊂ R and u (x , x ) := a(x − 1/4) + a(x − 1/4) with a(t) := t /6 + t/8 D 1 2 1 2 5 3 for t  0and a(t) := t /40 + t /8for t  0. Then the energy is given by (1.1) with β = 0 and f :=−divDW (Du ). The exact solution u = u satisfies the interface condition of D D Section ‘Refined analysis for an interface model problem’ and allows Theorem 5.3 to hold. Theorems 5.1 and 5.4 do not apply because β = 0. We use the grid of Figure 5 as initial triangulation to resolve discontinuities in ∇f . The algorithms of Subsection ‘Numerical algorithms’ ran with and without stabilisa- tion for uniform and adaptive meshes and led to Figure 6 with overall observations of Boiger and Carstensen Advanced Modeling and Simulation in Engineering Sciences 2013, 1:5 Page 16 of 23 http://www.amses-journal.com/content/1/1/5 Table 1 Observed convergence rates in Figures 3, 4, 6 and 7 for uniform and adaptive mesh refining 2 2 2 Example of subsection σ − σ  u − u  η η η D(u − u ) η R F L  H 2 2 L () L () L () unstab. stab. unstab. stab. unstab. stab. stab. unstab. stab. stab. unif 5/35/33/27/54/54/51 3/51/21/3 ‘Two-well benchmark’ R-adapt 2 7/5(5/3) 6/51 1 1 (2/3) 2/52/5 F-adapt 2 4/3(5/3) 6/51 1 1 (2/3) 2/52/5 unif 5/35/33/27/54/54/51 3/51/21/3 ‘Modified two-well benchmark’ R-adapt 2 7/5(5/3) 6/51 1 1 (2/3) 2/52/5 F-adapt 11/54/3(7/4) 6/51 1 1 (2/3) 2/52/5 unif (1) 3/2—7/514/51 — 1/22/5 ‘Three-well benchmark’ R-adapt 2 (1/4) — (1/4) 1 — (1/3) (1) (1/5) — F-adapt 9/51 — 4/513/54/5— 1/31/3 unif 4/54/56/52/5 ‘An optimal design example’ R-adapt 1 4/56/52/5 F-adapt 1 4/51 2/5 Convergence rates are given as powers of the representative mesh-size 1/ ndof which is proportional to H on uniform grids. Unavailable values are left blank, non-continuous rates are put in parantheses, inconclusive convergence behaviour is marked by “—”. Boiger and Carstensen Advanced Modeling and Simulation in Engineering Sciences 2013, 1:5 Page 17 of 23 http://www.amses-journal.com/content/1/1/5 Figure 4 Convergence plot of the modified two-well benchmark. Errors and error estimators of the modified two-well benchmark of Subsection ‘Modified two-well benchmark’, plotted against the number of degrees of freedom. Section ‘Conclusions’. Beyond those general conclusions, this example demonstrates the difficulties with ill-conditioned Hessians. While the unstabilised method reaches 10 degrees of freedom without difficulty on uniform meshes, the adapted algorithms fail without stabilisation beyond 687 324 degrees of freedom (η -adaptive) and 33 169 degrees of freedom (η -adaptive). MATLAB’s error message “Input to EIG must not contain NaN or Inf” indicates that a matrix operation returned non-finite numbers let fminunc break down. Undisplayed numerical experiments show condition numbers up to 10 and Figure 5 Initial grid for the three-well benchmark. Initial grid for the three-well benchmark of Subsection ‘Three-well benchmark’. Boiger and Carstensen Advanced Modeling and Simulation in Engineering Sciences 2013, 1:5 Page 18 of 23 http://www.amses-journal.com/content/1/1/5 Figure 6 Convergence plot of the three-well benchmark. Errors and error estimators of the three-well benchmark of Subsection ‘Three-well benchmark’, plotted against the number of degrees of freedom. beyond. The empirical convergence rates for uniform and R- as well as F-adapted mesh- refining are collected in Table 1. Moreover, Figure 1 in Section ‘Background’ reveals that stabilisation not only remedies ill-conditioned Hessians but thereby indeed allows for reduced errors in the discrete solution. An optimal design example The energy density of the topology optimisation problem of [3,8,29-33] reads W (F) := φ(|F|) for F ∈ R ⎪ t for 0  t  λ, √ √ √ √ with φ(t) := λ/2 + 2 λ(t − λ/2) for λ  t  2 λ, ⎪ √ t /2 + λ for t  2 λ. This leads to problem (2.3) with β = 0, λ = 0.0084, u ≡ 0and f ≡ 1. Since regu- larity of the solutions is unclear, only the results of Sections ‘Global convergence’, ‘A posteriori error estimates’, ‘Refined analysis for an interface model problem’ and ‘Numerical experiments’ apply. As initial triangulation T ,weuse thecoarsestcross triangulation T ={conv{(0, 0), (1, 0), (0, 1)},conv{(1, 0), (0, 1), (1, 1)}} of  = (0, 1) . The algorithms of Subsection ‘Numerical algorithms’ ran with and without stabilisa- tion for uniform and adaptive meshes and led to Figure 7 with the overall observations of Section ‘Conclusions’. The empirical convergence rates for uniform and R- as well as F-adapted mesh-refining are collected in Table 1. Undocumented experiments with a modified lower-order term f and known exact solution u led to the same convergence rates of the error estimators and confirm their accuracy. Boiger and Carstensen Advanced Modeling and Simulation in Engineering Sciences 2013, 1:5 Page 19 of 23 http://www.amses-journal.com/content/1/1/5 Figure 7 Convergence plot of the optimal design benchmark. Error estimators of the optimal design example of Subsection ‘An optimal design example’, plotted against the number of degrees of freedom. Discussion of Empirical Convergence Rates Global convergence without regularity assumptions Theorem 3.1 asserts that σ − σ  , βu − u  2 ,and |u | all tend to zero L () L () as H → 0. The plain convergence result applies to all examples from Subsections ‘Two-well benchmark’, ‘An optimal design example’, ‘Three-well benchmark’, and ‘An optimal design example’ for the uniform mesh-refinements with H = H /2. The numerical experiments, however, show empirical convergence rates displayed in the first columns of Table 1. The adaptive algorithms do not reflect the condition H → 0 explicitly and hence convergence is not guaranteed a priori. Undisplayed inves- tigations show that indeed in the R-adapted version of the three-well example of Subsection ‘Three-well benchmark’, this condition H → 0 does not appear to be true for more than 4 978 degrees of freedom. In all other experiments we observe convergence rates even for unstabilised discretisations. Empirical convergence rates for interface model problems Theorem 5.1 provides an a priori error estimate and an estimate of the stabil- isation norm. It applies to the benchmark of Subsections ‘Two-well benchmark’, ‘An optimal design example’, ‘Three-well benchmark’, and ‘An optimal design example’ only, because of β> 0 and Example 3.3, and the smoothness conditions imposed upon u from Section ‘Refined analysis for an interface model problem’. Recall the definitions of T ( ),  and  from Section ‘Refined analysis for an interface model problem’ −1 and assume u ≈ 1 ≈u , |T ( )|≈ H and | |≈ H in this dis- 2 C 2,p  , L (\ ) W ( ) 2/p cussion. This leads to a convergence rate of H for the right-hand side of Theorem 5.1. The observed convergence rates of σ − σ and u − u  2 for the stabilised L () L () Boiger and Carstensen Advanced Modeling and Simulation in Engineering Sciences 2013, 1:5 Page 20 of 23 http://www.amses-journal.com/content/1/1/5 benchmark examples in Table 1 show convergence rates beyond those guaranteed in Theorem 5.1. Theorem 5.3 implies, up to perturbations on the boundary, 1/3 −1/2 1/2 1/2 D(u − u )  u − u  +|u | + H |u | u − u  . 2     2 L () L () L () Since the exact solutions of Subsections ‘Two-well benchmark’, ‘An optimal design example’, ‘Three-well benchmark’, and ‘An optimal design example’ are all smooth up to a one-dimensional interface line, Theorem 5.3 applies to these examples. The experiments shows that the right-hand side of Theorem 5.3 is dominated by −1/2 1/2 1/2 H |u | u − u  in all examples and that the inequality is satisfied. L () Reliability without regularity assumptions Up to boundary terms, Theorem 4.1 states 2 2 σ − σ  + βu − u   η u − u  . 1,p 2 F p W () L () L () The convergence rates confirm this assertion for the general and rough estimate u − u  1,p  1 in the sense that the rates for η are worse than or equal to those W () 2 2 of σ − σ  and u − u  . In the numerical examples, u − u  1 is com- 2 H () L () L () puted and displayed in Table 1 and the convergence rates of the product u − u  1 η H () 2 2 can be compared with those of σ − σ  +u − u  . This comparison con- L () L () firms the above a posteriori error estimate. In the examples with p = 2 (of Subsections ‘Two-well benchmark’, ‘An optimal design example’, ‘Three-well benchmark’, and ‘An optimal design example’), there holds even equality of the convergence rates which demonstrates the efficiency of the estimate of Theorem 4.1. Efficiency without regularity assumptions Up to oscillations and the (possibly) higher-order term (id − I )σ , Theorem 4.2 F, q L () states η  σ − σ  + βu − u  . F  p  p L () L () The displayed convergence rates of Table 1 confirm this estimate. Reliability of the refined a posteriori error control Theorem 5.4 applies to the example of Subsection ‘Two-well benchmark’ and states 2 2 2 σ − σ  +u − u   η and D(u − u )  η . 2 L  2 H L () L () L () Table 1 confirms this estimate and shows that the estimators η and η accu- L H rately predict the convergence rate of the errors, even with equality of the conver- gence rates in the case of adaptive mesh refinements in the examples of Subsections ‘Two-well benchmark’, ‘An optimal design example’, ‘Three-well benchmark’, and ‘An optimal design example’. All displayed convergence rates of η arebetteroratleast equaltothose of η . For instance, for uniform mesh-refining in Subsections ‘Two-well benchmark’, ‘An optimal design example’, ‘Three-well benchmark’, and ‘An optimal design example’, 2 2 the error terms σ − σ  +u − u  converge with the empirical convergence L () L () rate 7/5 while the upper bound η does so with a reduced convergence rate 4/5. The F Boiger and Carstensen Advanced Modeling and Simulation in Engineering Sciences 2013, 1:5 Page 21 of 23 http://www.amses-journal.com/content/1/1/5 refined error estimator η is a guaranteed upper bound (via Theorem 5.4) and converges with an empirical convergence rate 1. Performance of the minimisation algorithm 1 In all numerical experiments of this paper, Algorithm 1 reaches the maximal number 3 of iterations. While this suggests that the optimal s is not found after three iterations, undisplayed experiments with higher iteration counts and hence higher computational efforts result solely in marginal improvements. Conclusions Effects of stabilisation The empirical convergence rates of the error estimators η , η and the errors F R u − u  2 and σ − σ  for uniform mesh-refinement with and without sta- L () L () bilisation coincide. This indicates that the choice γ = 1 leads to some significant perturbation but maintains the correct convergence rate at the same time. This is differ- ent for adaptive mesh refinement with less optimal convergence rates. Our conclusion is that an improved adaptive algorithm has to be developed with balance of local mesh- refinement and global stabilisation parameters in future research. The tested algorithm from Subsection ‘Numerical algorithms’ does neither reflect the effects of stabilisation nor that of inexact solve. Another important aspect of the stabilisation is the regularisation of the Hessian in the step SOLVE of Subsection ‘Numerical algorithms’. In the three-well problem of Subsection ‘Three-well benchmark’, the unstabilised adaptive algorithms fail. Adaptive versus uniform mesh-refinement The overall empirical convergence rates of the errors and estimators of the unstabilised computation for adaptive mesh-refinements are better than those for uniform mesh- refinements. This is in contrast to the stabilised computation, where the true errors σ − σ  and u − u  2 behave better for uniform compared with the two adap- L () L () tive mesh-refinments (with the exception in Subsection ‘An optimal design example’ where there is equality). It is observed that adaptivity does not necessarily improve the converegnce rates of the error σ − σ and u − u  2 in a stabilised compu- L () L () tation. Surprisingly, the convergence of the gradient errors D(u − u ) 2 are slightly L () improved in the instabilised calculation by adaptive mesh-refinements. The adaptive mesh-refinement is expected to reduce the a posteriori error estimators in the first place: cf. [1,34] for the estimator reduction property. Indeed, the convergence rates of the a posteriori error estimators η , η , η , η are improved (or optimal) for adap- R F L H tive mesh-refinements (except for the three-well example of Subsection ‘Three-well benchmark’). Strong convergence of the gradients The convergence of the gradient error of the stabilised problem surpasses the expecta- tions of [12] in Subsection ‘An optimal design example’ but fails to do so in Subsections ‘Two-well benchmark’, ‘An optimal design example’, ‘Three-well benchmark’, and ‘An optimal design example’. The improved error estimator η shows the same convergence rate as the error of the gradients in Subsections ‘Two-well benchmark’, Boiger and Carstensen Advanced Modeling and Simulation in Engineering Sciences 2013, 1:5 Page 22 of 23 http://www.amses-journal.com/content/1/1/5 ‘An optimal design example’, ‘Three-well benchmark’, and ‘An optimal design example’. This holds for uniform and for adapted mesh refinements and suggests that η is in fact reliable and efficient for β> 0. Guaranteed error control The assertion on η in Theorem 4.1 is reflected in the numerical examples in that the stress approximations converge faster than η in all cases. This suggests that the esti- mate u − u   1 is by far too pessimistic. In fact, the benchmark examples 1,p W () with known exact solution fulfil σ − σ   η u − u  . Similar affirmative 2 F H () L () conclusions follow for Theorem 4.2 and 5.4. Reliability-efficiency gap In comparison with the residual-based error estimator of [7,18], the new a posteriori error estimators η and η of Theorem 5.4 lead to refined error control. The improve- L H ment is marginal for uniform meshes without stabilisation but significant for adaptive stabilised computations. η and η match the convergence of the errors and so narrow L H the reliability-efficiency gap. Competing interests The authors declare that they have no competing interests. Authors’ contributions All authors contributed equally to all parts of this article. All authors read and approved the final manuscript. Received: 29 July 2013 Accepted: 6 December 2013 Published: 29 January 2014 References 1. Carstensen C (2008) Convergence of an adaptive fem for a class of degenerate convex minimisation problems. IMA J Numer Anal 28(3): 423–439 2. Dacorogna B (2008) Direct methods in the calculus of variations, 2nd Ed. Applied Mathematical Sciences 78. Springer, Berlin. xii 3. Carstensen C, Müller S (2002) Local stress regularity in scalar non-convex variational problems. SIAM J Math Anal 34(2): 495–509 4. Chipot M (2000) Elements of Nonlinear Analysis. Birkhäuser Advanced Texts. Basel, Birkhäuser. vi 5. Müller S (1999) Variational models for microstructure and phase transisions. In: Hildebrandt S, et al. (eds) Calculus of variations and geometric evolution problems. 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Studies in Mathematics and its Applications, vol. 8. North-Holland Publishing Co., Amsterdam. p 776 34. Cascón JM, Kreuzer C, Nochetto RH, Siebert KG (2008) Quasi-optimal convergence rate for an adaptive finite element method. SIAM J Numer Anal 46(5): 2524–2550 doi:10.1186/2213-7467-1-5 Cite this article as: Boiger and Carstensen: A posteriori error analysis of stabilised FEM for degenerate convex minimisation problems under weak regularity assumptions. Advanced Modeling and Simulation in Engineering Sciences 2013 1:5 Submit your manuscript to a journal and benefi t from: 7 Convenient online submission 7 Rigorous peer review 7 Immediate publication on acceptance 7 Open access: articles freely available online 7 High visibility within the fi eld 7 Retaining the copyright to your article Submit your next manuscript at 7 springeropen.com http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png "Advanced Modeling and Simulation in Engineering Sciences" Springer Journals

A posteriori error analysis of stabilised FEM for degenerate convex minimisation problems under weak regularity assumptions

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Abstract

cc@math.hu-berlin.de Department of Mathematics, Background: The discretisation of degenerate convex minimisation problems Humboldt-Universität zu Berlin, experiences numerical difficulties with a singular or nearly singular Hessian matrix. Unter den Linden 6, 10099, Berlin, Germany Methods: Some discrete analog of the surface energy in microstrucures is added to Department of Computational the energy functional to define a stabilisation technique. Science and Engineering, Yonsei University, Unter den Linden 6, Results: This paper proves (a) strong convergence of the stress even without any 120-749, Seoul, Korea smoothness assumption for a class of stabilised degenerate convex minimisation problems. Given the limitted a priori error control in those cases, the sharp a posteriori error control is of even higher relevance. This paper derives (b) guaranteed a posteriori error control via some equilibration technique which does not rely on the strict Galerkin orthogonality of the unperturbed problem. In the presence of L control in the original minimisation problem, some realistic model scenario with piecewise smooth exact solution allows for strong convergence of the gradients plus refined a posteriori error estimates. This paper presents (c) an improved a posteriori error control in this interface problem and so narrows the efficiency reliability gap. Conclusions: Numerical experiments illustrate the theoretical convergence rates for uniform and adaptive mesh-refinements and the improved a posteriori error control for four benchmark examples in the computational microstructures. Keywords: Adaptive finite element method; Relaxation; Convexification; Calculus of variations; Degenerate convex problems; Energy reduction; Nonconvex minimisation; Partial differential equation; Stabilisation; Strong convergence; A posteriori error estimate; Reliability-efficiency gap; Euler-Lagrange equation; Guaranteed upper bound Background Infimising sequences of variational problems with non-quasiconvex energy densities, in general, develop finer and finer oscillations with no classical limit in Sobolev func- tion spaces called microstructure [1-6]. Those oscillations cause difficulty to numerical methods because fine grids are necessary to resolve such oscillations which results in ineffective and tricky mesh-depending computations. Strong convergence of gradients of infimising sequences of the non-quasiconvex problem is impossible. Relaxation techniques replace the nonconvex energy density by its (semi-)convex hull and lead to a macroscopic model. Since the convexified energy density obtained by this © 2013 Boiger and Carstensen; licensee Springer. This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. Boiger and Carstensen Advanced Modeling and Simulation in Engineering Sciences 2013, 1:5 Page 2 of 23 http://www.amses-journal.com/content/1/1/5 method, in general, lacks strict convexity, numerical algorithms might encounter sit- uations where the Hessian matrix is singular. For instance, the Newton minimisation algorithm fails on the convexified three-well problem of Subsection ‘Three-well bench- mark’ below. Applications of relaxation techniques include models in computational microstructure [5-7], some optimal design problems [8,9], the nonlinear Laplacian [10] (where the Hessian can become arbitrarily ill-conditioned in spite of its strict convexity) and elastoplasticity [1]. Stabilisation techniques regularise the energy term by an additional positive semidefi- nite stabilisation function. The paper [11] discusses several choices of such stabilisation functions for P conforming finite elements and quasiuniform meshes. It turns out that stabilisation can ensure strong convergence of the strain approximations under particu- lar circumstances. A particular stabilisation in [12] leads to strong convergence even on unstructured grids but is still restricted to unrealistically smooth solutions. This paper studies the stabilisation technique of [12] and addresses the question of convergence (i) without extra regularity assumptions, (ii) in a realistic scenario called model interface problem, and (iii) establishes an a posteriori error control. The stabilisation leads to improved condition numbers of the Hessian matrix and to reduced errors if the numerical solvers fail without stabilisation. Figure 1 shows the con- vergence of the discrete stress σ of the three-well benchmark corresponding to the discrete minimisers of the energy E (v ) = E(v ) + C/2|v | . The errors are plot- ted for computations with uniform mesh refinements with various solver tolerances in the discrete minimisation procedure at a fixed triangulation and values of C,cf. Section ‘Numerical experiments’ for details on the MATLAB implementation. Without −5 stabilisation, the convergence stagnates with a moderate tolerance of 10 which becomes visible as a “plateau” in Figure 1. The Newton solver even aborts prematurely due to the singular Hessian. In conclusion, stabilisation enables higher accuracies in numerical examples. For β  0 the convex energy functional assumes the form E(v) := W (Dv(x)) + β|v(x) − g(x)| − f (x) · v(x) dx. (1.1) Assume that W is convex with quadratic growth so that there exist minimisers u ∈ H ();below p-th order growth is included while p = 2 throughout this simplifying Figure 1 Impact of the stabilisation on the error. Error of the stabilised stress σ with coefficients −4 −5 −6 −8 C = 0, 10 , 1 of the stabilisation and various tolerances tol = 10 ,10 ,10 of the Newton solver. Boiger and Carstensen Advanced Modeling and Simulation in Engineering Sciences 2013, 1:5 Page 3 of 23 http://www.amses-journal.com/content/1/1/5 introduction. Given a sequence of shape-regular triangulations (T ) [13], let u ∈N minimise the stabilised discrete energy 2 2 2 −1 2 E (v ) := E(v ) + |v | with |v | := H h || [Dv ] || F 2 L (F) F∈F () amongst all conforming P finite element functions v on T ,where [Dv ] is the jump of 1    F the gradient Dv along the interior side F,written F ∈ F (),and H := max h is the T T maximal diameter h of all simplices T ∈ T . Section ‘Global convergence’ verifies the strong convergence of the discrete solution u and its stress σ := DW (Du ) to their respective continuous conterparts, 2 2 2 ||σ − σ || + β||u − u || +|u | → 0as  →∞. 2  2 L () L () Section ‘A posteriori error estimates’ presents a novel application of [14-17] to non- linear problems. For the L projection  onto the space of piecewise P functions, any Raviart-Thomas function τ ∈ RT (T ) satisfies ||σ − σ || L () ||σ − τ || 2 +||  + div τ || 2 + osc ( ) ||u − u || 1 . ,2 L () L () H () This error bound holds for any discrete displacement u that satisfies the boundary con- ditions; the point is that inexact solve is included — there is no Galerkin orthogonality required. The drawback is to minimise the expression on the right-hand side with respect to τ in order to obtain a sharp error bound. This is a particular selection: degenerate con- vex minimisation problems do not allow for a control of ||u − u || 1 and may even face H () multiple exact or discrete solutions while the discrete minimum of E is unique. However, in some results of this paper, either W or the lower-order terms lead to some control over ||u − u || 2 and the selection via stabilisation is correct. H () Phase transition problems motivate the investigation of scenarios with a smooth solu- tion u up to a one-dimensional interface ⊂  [18]. Section ‘Refined analysis for an interface model problem’ proves that such problems allow even for strong convergence of 1,∞ 2 the gradients for any unique solution u in W ()∩H (\ ) [19]. This result also leads to an improvement of the a posteriori error control of the discrete stresses and narrows the efficiency-reliability gap; the efficiency-reliability gap is the difference of the conver- gence rates of the guaranteed upper a posteriori error bound and the guaranteed lower a posteriori error bound. Section ‘Numerical experiments’ complements the theoretical findings with numerical experiments to provide empirical evidence of the improved error control. The sta- bilisation technique competes in four benchmark examples, with and without known exact solution, for uniform and two different mesh-refining algorithms for the explicit residual-based error estimator of [7] and with an averaging-type error estimator of ([18], (1.11)). Standard notation on Lebesgue and Sobolev spaces is employed throughout this paper and a  b abbreviates a  Cb with some generic constant 0 < C < ∞ independent of crucial parameters (like the mesh-size on level ); a ≈ b means a  b  a.Furthermore, A : B abbreviates the matrix inner product that corresponds to the Frobenius norm. Boiger and Carstensen Advanced Modeling and Simulation in Engineering Sciences 2013, 1:5 Page 4 of 23 http://www.amses-journal.com/content/1/1/5 Methods: Discretisation and Stabilisation Based on the convergence results for unstructured grids, this paper will develop reliable error estimators for a class of stabilised convex minimisation problems described in the sequel. Let  ⊂ R be a bounded Lipshitz domain with polygonal boundary for n = 2or m×n 2 m 3. Given a continuous convex energy density W : R → R, g, f ∈ L (; R ), β  0, 1,p m and v ∈ W (; R ) with 2  p < ∞ and m = 1, ... , n, the energy is given by (1.1). 1 m×n Throughout this paper, the energy density W ∈ C (R ; R) satisfies (2.1)–(2.2) for parameters 1 < r  2, 0  s < ∞ and s + r + p  rp.The two-sided growth condition reads p p m×n |F| − 1  W (F)  |F| + 1 for all F ∈ R . (2.1) m×n The convexity control assumption reads, for all F , F ∈ R , 1 2 r s s |D W (F ) − D W (F )|  1 +|F | +|F | (D W (F ) − D W (F )) : (F − F ). 1 2 1 2 1 2 1 2 (2.2) The proof of Theorem 2 in [7] shows that (2.2) is crucial for the uniqueness of the stress tensor DW (Du). 2,p m 2 m Given Dirichlet data u ∈ W (; R ) ∩ H (∂; R ) for the set of admissible func- 1,p tions A := u + V := u + W (; R ), the continuous (convex) model problem D D reads minimise E(v) within v ∈ A. (2.3) A finite element approximation of (2.3) is based on a family of regular triangulations (T ) of the domain  into simplices in the sense of Ciarlet [13] (e.g., for n = 2, two ∈N non-disjoint triangles of T shareeitheracommonedgeoracommonnode).The setof sides F consists of edges (for n = 2) or faces (for n = 3) of T and is split into the union of the sets of all interiour sides F () and of all boundary sides F (∂). For latter reference, define the diameter h := diamT of a triangle (or tetrahedron) T ∈ T and the size h := diamF of a side F ∈ F .The mesh size function h :  → R F   >0 is given by h for x ∈ int T ∈ T , h (x) := min {h : F ∈ F and x ∈ F} otherwise. Theglobalmeshsizewillbeabbreviatedby H :=h  ∞ .Wepresume thefamily L () (T ) to be shape-regular so that h ≈ h for all T ∈ T , F ∈ F and F ⊂ T. ∈N F T The space of T -piecewise polynomials of degree  k ∈ N is P (T ).The nodal 0 k interpolation I w ∈ P (T ) ∩ C() of w ∈ C() is given by I w(z) = w(z) for all 2 2 nodes z.Let furthermore  w be the L projection of w ∈ L () onto P (T ),and osc (w) :=h (id −  )w be the oscillation of w ∈ L () for 2  q  ∞ ,q   L () with respect to the triangulation T . The symbol id denotes the identity operator. Let u = I u ,and D,  D A := u + V with V := V ∩ P (T ; R ) ∩ C(). D,   1 Given a function v on  which is possibly discontinuous along some side F ∈ F () shared by the two elements T such that there exist traces from either sides, the jump of v along F reads Boiger and Carstensen Advanced Modeling and Simulation in Engineering Sciences 2013, 1:5 Page 5 of 23 http://www.amses-journal.com/content/1/1/5 [v] (x) =[v] (x) := lim v( y) − lim v( y) for x ∈ F. T y→x T y→x + − The stabilisation of [12] will be used throughout this paper with −1 <γ < ∞ and 1+γ a (v, w) := [Dv] :[Dw] ds and |v| := a (v, v). (2.4) F F F F F∈F () The stabilised discrete problem reads minimise E (v) := E(v) + a (v, v) amongst v ∈ A . (2.5) Convergence of gradients with a guaranteed convergence rate is shown in [12] under unrealistically high regularity assumptions. A comprehensive collection of the results in [12] is summarised in the following theorem. 3/2+ε m Theorem 2.1. ([12])Let u ∈ A ∩ H (; R ) be some solution of (2.3) for some ε> 0; let p and r be the Hölder conjugate of p and r, −1 <γ < 3, and set ζ := min 1 + γ , r for β> 0 and ζ := min {1 + γ ,2} for β = 0. Then the discrete solution u ∈ A of (2.5) and the continuous and discrete stress σ := p m×n m×n (; R ) and σ := DW (Du ) ∈ P (T ; R ) satisfy DW (Du) ∈ L (1+γ)/2 ζ r 2 2 2 σ − σ  +u − u  + |||u ||| + H D(u − u )  H . 2   2 L ()  L () L () Proof. This combines Lemma 3.5 and 4.1–4.2 plus Theorem 3.8 and 4.4 in [12]. Global convergence This section is devoted to the proof of a general convergence result without higher reg- ularity assumptions. Let u ∈ A and u ∈ A solve the minimisation problem (2.3) and (2.5) and set σ := DW (Du) and σ := DW (Du ). For the unstabilised approximation, the a priori error estimates of [7] plus a density argument prove convergence of r 2 σ − σ  + βu − u  → 0as H → 0. L () L () Note that β = 0 is permitted. Then, however, uniqueness of u and convergence of u − u  are guaranteed. The point in the following result is that the stabilised approxi- L () mation converges as well as |||u ||| → 0 even for non-smooth or non-unique minimisers. Under special circumstances, uniqueness of u and the convergence u − u  2 → 0 L () can be shown even for β = 0, e.g., in Example 3.3. Theorem 3.1. (Global Convergence)Provided lim H = 0 it holds →∞ r 2 2 σ − σ  + βu − u  + |||u ||| → 0 as  →∞. L () L () The proof is based on the following lemma. Lemma 3.2. The errors δ := σ − σ and e := u − u satisfy, for all v ∈ V ,that r 2 r 2 δ  + βe   |e − v | + βe − v | + a (u , v ). 2   1,p   2 L () W () L () L () Proof. The minimisation problems (2.3) and (2.5) are equivalent to their respective Euler-Lagrange equations, namely for v ∈ V and v ∈ V , σ(x) :Dv(x) + 2β(u(x) − g(x)) · v(x) − f (x) · v(x) dx = 0; (3.1) Boiger and Carstensen Advanced Modeling and Simulation in Engineering Sciences 2013, 1:5 Page 6 of 23 http://www.amses-journal.com/content/1/1/5 σ (x) :Dv (x) + 2β(u (x) − g(x)) · v (x) − f (x) · v (x) dx + a (u , v ) = 0. (3.2) Algebraic transformations of the difference of these two equations lead to ˆ ˆ δ :De dx + 2βe  = (δ :D(e − v ) + 2βe · (e − v ))dx + a (u , v ). L () It is shown in ([12], Lemma 3.5) that δ   δ :De dx. (3.3) L () Two Hölder inequalities on the right-hand side and absorbtions of δ and L () e  2 eventually concludethe proof. Furtherdetails aredropped forbrevity. L () Proof of Theorem 3.1. Given any positive ε, the density of smooth functions in 1,p m m W (; R ) leads to some v ∈ D(; R ) such that u − u − v   ε.Hence 1,p ε D ε 0 W () v := I (v + u ) − u ∈ V satisfies ε D e − v = (u − u − v ) + (id − I )(v + u ). D ε  ε D Note that the nodal interpolation I (v + u ) is well-defined since v and u are assumed ε D ε D to be smooth. With ([12], Lemma 3.1–3.2) it follows that (id − I )(v + u ) 1,p  H → 0and ε D W () 1+γ 2 2 |||I (v + u )||| = |||(id − I )(v + u )|||  H → 0as  →∞. ε D  ε D Since · 2  · 1,p , this yields some  ∈ N such that L () W () r 2 2 |e − v | + βe − v  () + |||I (v + u )|  ε for all    . 1,p   2  ε D 0 W () L A Cauchy inequality applied to the stabilisation norm proves 1 1 2 2 2 a (u , v ) = −|||u ||| + a (u , I (v + u ))  − |u | + |I (v + u )| . ε D   ε D 2 2 Substitute a (u , v ) in Lemma 3.2 and add |||u ||| on both sides. This leads to r 2 2 δ  + βe  + |||u |||  ε for all    . 2  0 L () L () Example 3.3. The two-well example from the computational benchmark [18] allows an estimate on e  2 even for β = 0. Let n = 2, let F :=−F := (3, 2)/ 13, and let the L () 1 2 2 2 energy density W be theconvexhull of F →|F − F | |F − F | .Thatis 1 2 2 2 2 W (F) = max 0, |F| − 1 + 4 |F| − (3F(1) + 2F(2)) /13 . (3.4) Then ([11], Lemma 9.1) proves, for all v ∈ V ,that 2 2 e   δ :De dx +e − v  . 2     1 L () H () Therefore, the arguments of Lemma 3.2 lead to r 2 r 2 δ  +e   |e − v | +e − v  + a (u , v ). 2   1,p   1 L () W () H () L () This result can be used in the proof of Theorem 3.1 in order to obtain r 2 2 σ − σ  +u − u  + |||u ||| → 0as  →∞. L () L () Boiger and Carstensen Advanced Modeling and Simulation in Engineering Sciences 2013, 1:5 Page 7 of 23 http://www.amses-journal.com/content/1/1/5 A posteriori error estimates Beyond the a posteriori error analysis of [7], the additional stabilisation term in the dis- cretisation of this paper causes an additional difficulty in that the Galerkin orthogonality does not hold for the natural residual. Inspired from novell developments in the a posteri- ori error control of elliptic PDEs motivated by inexact solve [14-17], this section presents some guaranteed upper error bound for the discretisation at hand for any approximation u which does not necessarily satisfy (3.2) exactly. Thereby inexact solve is included. Let u ∈ A solve (2.3) and let u ∈ A be arbitrary. It is not assumed that u solves the discrete problem (2.5); the following theorem holds regardless of this. Recall the def- initions of osc (·) and  from Section ‘Methods: Discretisation and Stabilisation’ and ,q given σ := DW (Du) and σ := DW (Du ), abbreviate :=−2β(u − g) + f , e := u − u and δ := σ − σ . 1,p m Theorem 4.1. Given any w ∈ W (; R ) with w = u − u on the boundary ∂,and m×n given any τ ∈ H(div, ; R ), it holds, for all 2  q  p and for some constant κ known from ([12], Lemma 3.5), that r 2 1−r r 2 κ/2δ  + βe   (rκ/2) /r |w | + βw 2  1,p  2 L () W () L () L () + σ − τ  +  + divτ  + osc ( ) e − w  1,q . q   q ,q W () L () L () The constant κ depends on problem-specific data such as u 1,p and the size of the W () domain . Refer to the proof of Lemma 3.5 in [12] for details. Before the proofs conclude this section, some practical choice of τ in Theorem 4.1 is discussed as some Raviart-Thomas finite element functions in RT (T ) := τ ∈ P (T )∩H(div, ) : ∀T ∈ T ∃a, b, c ∈ R ∀x ∈ T, τ (x) = (a, b)+cx . 0  RT 1   RT We suggest the computation (or an accurate approximation) of μ := min σ − τ +  + divτ (4.1) q q L () L () τ ∈RT (T ) and emphasise that any upper bound is allowed in Theorem 4.1. This leads to r 2 1−r r 2 κ/2δ  + βe   (rκ/2) /r |w | + βw 2 1,p 2 L () W () L () L () + μ + osc ( ) e − w  1,q . ,q W () The algorithm of ([20], Prop. 4.1) computes some w from (id − I )u with 1/q q q w  ≈ h (id − I )u  and (4.2) L (T ) T  D L (∂T ∩∂) 1/q−1 1/q q q q Dw   h (id − I )u  + h ∂(id − I )u /∂s . L (T )  D L (∂T ∩∂)  D L (∂T ∩∂) T T (The proof of the second assertion is analogous to that of ([20], Prop. 4.1) and the first is an immediate consequence of the design of w ). This and e − w  1,q  1for W () bounded u (i.e. solely u  1,p  1 is assumed) lead to the practical estimate μ as a W () computable guaranteed upper bound of the left-hand side of Theorem 4.1. Since the min- imisation of (4.1) is computationally intensive for q = 2, Section ‘Numerical experiments’ actually computes an approximation of μ ,based on q = 2. The choice τ = σ in Theorem 4.1 shows that the right-hand side is in fact optimal up to oscillations.The reliability-efficiency gapof[18] is visiblehereinthatwehavenofurther estimate on u  1,p [7,18]. However, additional smoothness assumptions on u may W () lead to refined estimates on the term e − w  1,q (cf. Section ‘Refined analysis for W () Boiger and Carstensen Advanced Modeling and Simulation in Engineering Sciences 2013, 1:5 Page 8 of 23 http://www.amses-journal.com/content/1/1/5 an interface model problem’). The following result indicates that μ is sharp in the sense that it converges with the correct convergence rate. This theorem employs the Fortin t n interpolation operator I defined for τ ∈ H(div, ) ∩ L (; R ) with t > 2by I τ ∈ F, F, RT (T ) and n · (id − I )τds = 0 for all F ∈ F . F F, Here and in the following, n denotes a unit normal vector of the side F; the direction of n arbitrary, but fixed for a given side F. For the improved regularity of stress in the class of degenerate convex minimisation problems at hand, we refer to [3,21]. Theorem 4.2. (Efficiency)Ifthe exactstress σ is sufficiently regular such that its Fortin m×n interpolant τ = I σ ∈ RT (T ; R ) is defined, it holds F, 0 σ − τ +  + divτ q    q L () L () δ  + 2βe  +(id − I )σ  . q  q F, q L  L () L () It is expected that (id − I )σ   H . This is shown in ([22], Prop. 3.6) for F, q L () q = 2 and therefore also holds for q  2. Hence the right-hand side of the assertion of Theorem 4.2 converges with the (expected) optimal convergence rates. Proof of Theorem 4.1. Let κ be the reciprocal of c in ([12], Lemma 3.5), which is also the multiplicative constant hidden in (3.3). Recall Young’s inequality, which reads ab r r a /r + b /r for a, b > 0. This, (3.3) and the continuous Euler-Lagrange equation (3.1) show, for v = e − w ∈ V,that r 2 κδ  + 2βe   (δ :Dv + 2βe · v)dx L () L () + (δ :Dw + 2βe · w )dx − (σ :Dv −  · v)dx 2 2 + βe  + βw 2  2 L () L () r 1−r + κ/2δ  + rκ/2 /r |w | . ( ) 1,p W () L () Hence Res (v) :=− (σ :Dv −  · v)dx satisfies r 2 1−r r 2 κ/2δ  + βe   Res (v) + (rκ/2) /r |w | + βw  . 2 1,p 2 L () W () L () L () 1,q Let C denote the Poincaré constant of convex domains with respect to the W norm. The fundamental theorem of calculus on some one-dimensional arc shows that C  1. The paper [23] proves C = 1/2. Hence, operator-interpolation arguments [24,25] prove 1/q C  (1/2)  1. The Poincaré inequality shows, for any 2  q  p,that ˆ ˆ (id −  ) · vdx = h (id −  ) · (id −  )vdx h (id −  ) Dv q = osc ( )Dv q . q L () ,q  L () L () m×n For any τ ∈ H(div, ; R ), the Hölder and Poincaré inequalities show Res (v) =− (σ − τ) :Dv − (  + divτ) · v − (id −  ) · v dx σ − τ  +  + divτ  + osc ( ) v 1,q . q   q ,q W () L () L () Boiger and Carstensen Advanced Modeling and Simulation in Engineering Sciences 2013, 1:5 Page 9 of 23 http://www.amses-journal.com/content/1/1/5 Proof of Theorem 4.2. The triangle inequality yields σ − τ (id − I )σ +δ q q q F, L () L () L Since f = 2β(u − g) − divσ , the commutative property divI =  div (cf. ([22], p. 129)) F, yields + divτ  = 2β e   2βe  . q   q  q L () L () L () Refined analysis for an interface model problem This section is devoted for a model scenario from phase transition problems [18] with some solution u that is smooth outside some one-dimensional interface .Sup- pose some (possibly non-unique) minimiser u of the continuous problem (2.3) satisfies 1,∞ m 2,p m u ∈ W (; R ) ∩ W ( \ ; R ) for some finite union of (n − 1) dimensional Lipschitz surfaces in .Since  has a Lipschitz boundary, this implies Lipschitz continu- 1,∞ m ity of u on . We refer to [19] for sufficient conditions for u ∈ W (; R ) and conclude 2,p m that the remaining assumption u ∈ W ( \ ; R ) is the essential hypothesis expected in many interface problems. Let u ∈ A be the(unique)minimiser of thediscretesta- bilised problem (2.5). In the following, also =∅ is permittedtoextendpreviousresults [12] for highly regular minimisers. The following theorem leads to a priori convergence rates for the interface model prob- lem. Thereby it recovers the results of [12] for problems with piecewise smooth exact solution. We will abbreviate the set of all triangles that are touched by as T ( ) :={T ∈ T : dist(T, ) = 0}, its cardinality as |T ( )|,its unionas  := int( T ( )) with volume | | and its complement as  :=  \  . , , Theorem 5.1. Provided β> 0,itholds 1+γ r/(r−1) r/(r−1) r 2 2 2 2 2 δ  +e  +|u |  H |u| + H |u| +H |u| 2  2 1,∞ p   2,p C L ()  H (\ ) W () L () W ( ) γ +n−1 r/(r−1) 2 r/((r−1)p) + H |u| |T ( )|+|u| | | . 1,∞ 1,∞ W () W () 1−n Remark 5.2. In the case of uniform mesh refinements we may expect |T ( )|≈ H and | |≈ H and Theorem 5.1 simplifies to r/((r−1)p) min {γ ,2} r/(r−1) r 2 2 2 δ  +e  +|u |  H |u| + H |u| . 2  1,∞ p 1,∞ L () W () W () L () Proof. With w = (id − I )e = (id − I )u, a Young inequality, (3.3) and ([12], Theorem 3.8) yield r/(r−1) r 2 2 2 2 δ  +e  +|u |  |w | +w  +|I u| . 2   1,p  2 L () W () L () L () Theorem 4.4.4 in [25] shows w  2  w  ∞  H |u| 1,∞ and L () L () W () p p p |w | =|w | +|w | 1,p  1,p 1,p C W () W ( ) , W ( ) p p p |u| | |+ H |u| . 1,∞ C 2,p W ( ) , W ( ) , Boiger and Carstensen Advanced Modeling and Simulation in Engineering Sciences 2013, 1:5 Page 10 of 23 http://www.amses-journal.com/content/1/1/5 Let ω = T be the patch of a side F ∈ F ,and set F ( )= {F ∈ F () : ω ∩ = F    F T ∈T F⊂T C C ∅} and F ( ) = F () \ F ( ).Notethat[Du] =0for F ∈ F ( ).Then ⎛ ⎞ 1+γ ⎜ ⎟ 2 −1 2 −1 2 |I u| = H h [Dw ]  + h [DI u]  · ⎝ ⎠ F 2  F 2 F L (F) F L (F) F∈F ( ) F∈F ( ) The first sum can be estimated as in the proof of ([12], Lemma 3.2), the second sum with 2 n−1 2 n−1 2 [DI u]   h |I u|  h |u| . F 2  1,∞ 1,∞ L (F) F W (F) F W (F) The observation |F ( )|  (n + 1)|T ( )| concludes the proof. Together with Theorem 5.1, the subsequent result implies strong convergence of the gradients in the model interface problem as H → 0. Theorem 5.3. Under the aforementioned conditions on the (possibly non-unique) exact 1,∞ m 2,p m minimiser u ∈ W (; R ) ∩ W ( \ ; R ),the errore = u − u of the discrete solution u ∈ A of (2.5) satisfies 1/3 5/6 1/3 (1−γ)/2 2 2 De  2 e  + H ∂ u /∂s  + H |u | L ()  2 D 2 L () L (∂) −(1+γ)/4 1/2 1/2 5/4 1/2 2 2 + H |u | e  + H ∂ u /∂s  . 2 2 L () L () Proof. The basic idea of gradient control is the generalisation of the interpolation esti- 1 2 2 mate H () = [L (), H ()] for a reduced domain \ ; refer to [24,25] for a detailed 1/2 analysis of interpolation spaces. Let w be the boundary value interpolation of (id − I )u as described in ([20], Prop. 4.1), such that w satisfies the inequalities in (4.2). A piecewise 1,p integration by parts shows, for v := e − w ∈ W (; R ),that ˆ ˆ De  = D(u − u ) :Dvdx + De :Dw dx L () ˆ ˆ ˆ v · [Du] n ds − v · u dx − v · [Du ] n ds F F \ F F∈F () +De  2 Dw  2 , L () L () where n is a unit normal vector of the interface . The Lipschitz continuity of u implies |[Du] n |  1. This and the trace inequality on lead to 1/2 1/2 v · [Du] n ds  v 2  v 2 +v Dv . L ( ) L () 2 2 L () L () The case = ∅ is contained in ([12], Theorem 4.4). The piecewise Laplacian of u is bounded in L () and so (with the generic constant C :=u 2 hidden in the L (\ ) notation C ≈ 1) v · udx  v 2 L () The elementwise trace inequality ([25], Theorem 1.6.6, p. 39) for an n-dimensional 1,q m simplex T and one of its sides F,and f ∈ W (T; R ),1  q < ∞,reads q q q−1 q q−1 q −1 −1 f   h f  +f  Df  q  h f  + h Df  . q q q q q L (T ) L (F) T L (T ) L (T ) T L (T ) T L (T ) Boiger and Carstensen Advanced Modeling and Simulation in Engineering Sciences 2013, 1:5 Page 11 of 23 http://www.amses-journal.com/content/1/1/5 The term v · [Du ] n ds and the stabilisation |u | are already analysed in the Estimate on C in the proof of ([12], Theorem 4.4). This results in (1−γ)/2 −(1+γ)/2 v · [Du ] n ds  |u | H Dv 2 + H v 2 . F L () L () F∈F () The preceding estimates plus the absorbtion of De  2 lead to L () 1/2 1/2 2 2 De   v +v Dv +Dw 2 2 2  2 L () L () L () L () L () (1−γ)/2 −(1+γ)/2 +|u | H Dv 2 + H v 2 . L () L () The triangle inequality applied to v = e − w and some careful elementary analysis to 1/2 absorb De  eventually lead to L () 1/3 1/3 (1−γ)/2 De  2  e  +w  +|w | 1 + H |u | L ()  2  2  H () L () L () 1/2 −(1+γ)/4 1/2 + H |u | e  2 +w  2 . L () L () The inequalities (4.2), Poincaré and Friedrichs inequalities on sides F ∈ F (∂) and removal of higher-order terms in H conclude the proof. The following theorem is an improved a posteriori estimate based on Theorems 4.1 and 5.3. 1,∞ m 2,p m Theorem 5.4. Recall u ∈ W (; R ) ∩ W ( \ ; R ), the definitions e := u − u and δ := σ − σ for σ := DW (Du) and σ := DW (Du ), and the definition of  from Section ‘A posteriori error estimates’. Set M(τ ) :=σ − τ  2 +  + divτ  2 + osc ( ) ,2 L () L () m×n for all τ ∈ H(div, ; R ) . Provided β> 0,itholds −(1+γ)/3 2/3 r 2 6/5 4/3 δ  +e   M(τ ) + H M(τ ) |u | L () L () (1−γ)/2 1−γ/4 1/2 min {5,r (1+1/p)} + M(τ ) H |u | +H |u | +H and min {5/3,r (1+1/p)/3} −(1+γ)/9 2/9 2 2/5 4/9 De   M(τ ) + H M(τ ) |u | + H L () 1/3 (1−γ)/2 1−γ/4 1/2 1−γ 1/3 2 + M(τ ) H |u | + H |u | + H |u | 1/2 min 5,r (1+1/p) −(1+γ)/2 −(1+γ)/3 2/3 { } 6/5 4/3 + H |u | M(τ ) + H M(τ ) |u | + H 1/2 −(1+γ)/2 (1−γ)/2 1−γ/4 1/2 1/2 + H |u | M(τ ) H |u | + H |u | The generic constants in Theorem 5.4 depend on problem-specific data such as the shapes of  and as well as the generic constant κ of Theorem 4.1. Theorem 5.5. Theorem 5.4 holds verbatim in Example 3.3 and in the modified two-well problem of Subsection ‘Modified two-well benchmark’, where β = 0. Remark 5.6. The assertion of Theorem 5.4 holds for any discrete u ∈ u + V which D, may approximate the discrete unique exact solution of (2.5). This allows the inexact SOLVE via an iterative procedure. Boiger and Carstensen Advanced Modeling and Simulation in Engineering Sciences 2013, 1:5 Page 12 of 23 http://www.amses-journal.com/content/1/1/5 Proof of Theorem 5.4. Choose w as in the proof of Theorem 5.3. Then Theorem 4.1 with q = 2 and (4.2) imply r 2 r 2 δ  +e   M(τ )e − w  1 +|w | +w 2    1,p  2 p H () L () W () L () L () min 5,r (1+1/p) 3/2 { } M(τ ) |e | +e  + H + H . 1 2 H () L () Theorem 5.3 provides an estimate of the semi-norm |e | 1 . A Young inequality shows H () 5/6 5 6 H M(τ )  H +M(τ ) . The absorbtion of e  2 then proves the first assertion. The L () second assertion is an immediate consequence of the first one, Theorem 5.3 and several algebraic transformations. Numerical experiments This section illustrates the theoretical estimates and their impact on the reliability- efficiency gap on 2D benchmarks in computational microstructures [18,26]. Numerical algorithms The adaptive finite element method (AFEM) and algorithmic details on the implementa- tion in MATLAB in the spirit of [27] concern the state-of-the-art AFEM loop SOLVE → ESTIMATE → MARK → REFINE and are explained below together with some notation. Solve The stabilised discrete problem (2.5) is solved in a nested iteration on a given triangulation T with MATLAB’s standard-minimiser fminunc with default tolerances. Gradient and Hessian of the discrete energy are available and therefore provided to fminunc.Weset γ = 1 in the stabilisation term (2.4) in all our experiments. This is motivated by ([12], Theorem 4.4) which suggest that γ = 1 yields an optimal convergence rate. The discrete solution of the previous AFEM loop iteration serves as a start vector for fminunc;for the first iteration, the initial vector is zero everywhere up to the Dirichlet boundary nodes. Since the Galerkin orthogonality is not required in Theorem 4.1, the termination of an iterative realisation for SOLVE is not a sensitive issue. In the computational PDEs, it is a fundamental issue to involve inexact solve. In this paper, however, the numerical examples are run with the standard settings of MATLAB. Estimate The refinement indicator results from the error estimator of Theorem 4.1. As in the work of Repin [28], the computation of the minimiser τ ∈ RT (T ) of σ − τ  2 +  + divτ  2 (6.1) L () L () runs Algorithm 1 based on the formula 2 2 2 (a + b) = min (1 + s)a + (1 + 1/s)b for a, b > 0 s>0 The stopping criterion of Algorithm 1 monitors relative changes and avoids degenerate values of s. Undisplayed experiments have conviced us that a maxmium of three iterations 0.8 and a stopping tolerance of ε (with the machine precision ε ) yield satisfying results. M Boiger and Carstensen Advanced Modeling and Simulation in Engineering Sciences 2013, 1:5 Page 13 of 23 http://www.amses-journal.com/content/1/1/5 Algorithm 1 Approximate flux computation Input: σ , s = 1; for k = 1, 2, 3 do Compute minimiser τ of 2 2 M(s , τ) = (1 + s )σ − τ  + (1 + 1/s )  + divτ  ; k k 2 k 2 L () L () if D (s , τ ) nearly singular (MATLAB “warning”) then return τ ; k k k s =  + divτ  /σ − τ  ; 2 2 k+1   k  k L () L () |s −s | k+1 k 0.8 if max s ,1/s , <ε then return τ ; k+1 k+1 k s +s M k+1 k Output: approximate flux τ The iteration is stopped whenever s,1/s or the relative change of s drops below this toler- ance. As an additional precaution, the iteration also stops if the linear system is deemed “nearly singular” by MATLAB. Our experiments convinced us that ignoring this warn- ing causes a breakdown with NaNs. Note that if q = 2, we still minimise the L sums in (6.1) to avoid the computational cost of a nonlinear solve. With the computed minimiser τ, Section ‘A posteriori error estimates’ yields the error estimator η :=σ − τ  +  + divτ  + osc ( ) . F,q  q   q ,q L () L () This will be compared with the well-established residual based a posteriori error estimator [7] ⎛ ⎞ ⎛ ⎞ 1/q 1/q q q q ⎝ ⎠ ⎝ ⎠ η := h   + h [σ ] · n  , R,q F  F q T q L (F) L (T ) T ∈T F∈F () which is reliable for the original discretisation without stabilisation. Undisplayed exper- iments computed the averaging error estimator [18], which is founded on the same theoretical background as η and therefore yielded essentially the same convergence R,q rates. The error estimators in Theorem 5.4 read 6/5 −(1+γ)/3 4/3 2/3 η := η + H η |u | L,2 F,2  F,2 (1−γ)/2 1−γ/4 1/2 min {5,r (1+1/p)} + η H |u | + H |u | + H F,2 min 5/3,r (1+1/p)/3 2/5 −(1+γ)/9 4/9 2/9 { } η := η + H η |u | + H H,2 F,2 F,2 1/3 1/3 (1−γ)/2 1−γ/4 1/2 1−γ + η H |u | + H |u | + H |u | F,2 1/2 min 5,r (1+1/p) −(1+γ)/2 6/5 −(1+γ)/3 4/3 2/3 { } + H |u | η + H η |u | + H F,2  F,2 1/2 −(1+γ)/2 (1−γ)/2 1−γ/4 1/2 1/2 + H |u | η H |u | + H |u | . F,2    Boiger and Carstensen Advanced Modeling and Simulation in Engineering Sciences 2013, 1:5 Page 14 of 23 http://www.amses-journal.com/content/1/1/5 MARK For any given T ∈ T with its set of faces F (T ), ∂T = F (T ),and given τ from (6.1), set q q q q η (T ) :=σ − τ  +  + divτ  + h (id −  )  . F q q q L (T ) L (T ) L (T ) q q q q /n 1/n η (T ) :=|T |   +|T | [σ ] ·n  . F F R q q L (T ) L (F) F∈F ()∩F (T ) q q Let η (T ) be one of the refinement indicators η (T ) and η (T ). Some greedy algorithm computes M ⊂ T of (almost) minimal cardinality such that q q η (T )  1/2 η (T ). T ∈M T ∈T Refine This step computes the smallest refinement T of T with M ⊂ T \ T based on +1    +1 the red-green-blue refinement strategy as illustrated in Figure 2. This refinement involves some closure algorithm to avoid hanging nodes. Two-well benchmark The computational microstructure benchmark of ([18], Section 2) considers two wells with W from (3.4) in Example 3.3. The energy is given by (1.1) on the domain  = (0, 1)× (0, 3/2) ⊂ R with g(x) for t  0, 5 3 g(x) :=−3t /128 − t /3and u (x) := ⎩ 3 t /24 + t for t  0 for t := (3(x − 1) + 2x )/ 13; p = q = 4and f ≡ 0. The unique minimiser u of 1 2 1,4 min E(v) with A = u + W () reads u = u ([18], Theorem 2.1) and β = 1 allows v∈A D D for Theorems 5.1–5.4 to hold. An initial triangulation T is given by a criss triangula- tion of (0, 1) × (0, 3/2) with 12 congruent triangles and the two interior nodes (1/2, 1/2) and (1/2, 1). The adaptive algorithm of Subsection ‘Numerical algorithms’ computes a sequence of discrete solutions (u ) and stresses (σ ) ,aswellaserror estimators η and η with and without stabilisation for uniform and adaptive meshes and led to Figure 3 with overall observations of Section ‘Conclusions’. The empirical convergence rates for uniform and R- as well as F-adapted mesh-refining are collected in Table 1. Note that the error estimator η performs better than η . This is evident from the table for uniform L F mesh refinements, but a closer look at Figure 3 reveals that even in the adaptive scenarios, Figure 2 Possible refinements of a triangle T. Boiger and Carstensen Advanced Modeling and Simulation in Engineering Sciences 2013, 1:5 Page 15 of 23 http://www.amses-journal.com/content/1/1/5 Figure 3 Convergence plot of the two-well benchmark. Errors and error estimators of the two-well benchmark of Subsection ‘Two-well benchmark’, plotted against the number of degrees of freedom. η converges slightly faster than η .Thisisinaccordance to thetheoryofSection L F ‘Refined analysis for an interface model problem’ where η is derived from η based on L F additional smoothness assumptions. Modified two-well benchmark This subsection concerns a modification of the previous problem with (3.4) and a linear right-hand side for β = 0and f (x) :=−div(DW (Du (x))) and unique solution u = u D D as before. Note that Example 3.3 applies to this problem, and so the proof of Theorem 3.1 yields r 2 2 σ − σ  +u − u  +|u | → 0as  →∞ L () L () and Theorems 5.1–5.4 hold as well. The algorithms of Subsection ‘Numerical algorithms’ ran with and without stabilisation for uniform and adaptive meshes with the same initial triangulation as in Subsection ‘Two-well benchmark’ and led to Figure 4 with overall observations of Section ‘Conclusions’. The empirical convergence rates for uniform and R- as well as F-adapted mesh-refining are collected in Table 1 for completeness although they are almost identical with those observed in Subsection ‘Two-well benchmark’. Three-well benchmark The energy density W of ([26], Example 5.9.3, p. 72) is the convex hull of min{|F| , |F − 2 2 (1, 0)| , |F − (0, 1)| } with explicit form in ([26], Example 5.6.4, p. 58). Let furthermore 2 2 3 = (0, 1) ⊂ R and u (x , x ) := a(x − 1/4) + a(x − 1/4) with a(t) := t /6 + t/8 D 1 2 1 2 5 3 for t  0and a(t) := t /40 + t /8for t  0. Then the energy is given by (1.1) with β = 0 and f :=−divDW (Du ). The exact solution u = u satisfies the interface condition of D D Section ‘Refined analysis for an interface model problem’ and allows Theorem 5.3 to hold. Theorems 5.1 and 5.4 do not apply because β = 0. We use the grid of Figure 5 as initial triangulation to resolve discontinuities in ∇f . The algorithms of Subsection ‘Numerical algorithms’ ran with and without stabilisa- tion for uniform and adaptive meshes and led to Figure 6 with overall observations of Boiger and Carstensen Advanced Modeling and Simulation in Engineering Sciences 2013, 1:5 Page 16 of 23 http://www.amses-journal.com/content/1/1/5 Table 1 Observed convergence rates in Figures 3, 4, 6 and 7 for uniform and adaptive mesh refining 2 2 2 Example of subsection σ − σ  u − u  η η η D(u − u ) η R F L  H 2 2 L () L () L () unstab. stab. unstab. stab. unstab. stab. stab. unstab. stab. stab. unif 5/35/33/27/54/54/51 3/51/21/3 ‘Two-well benchmark’ R-adapt 2 7/5(5/3) 6/51 1 1 (2/3) 2/52/5 F-adapt 2 4/3(5/3) 6/51 1 1 (2/3) 2/52/5 unif 5/35/33/27/54/54/51 3/51/21/3 ‘Modified two-well benchmark’ R-adapt 2 7/5(5/3) 6/51 1 1 (2/3) 2/52/5 F-adapt 11/54/3(7/4) 6/51 1 1 (2/3) 2/52/5 unif (1) 3/2—7/514/51 — 1/22/5 ‘Three-well benchmark’ R-adapt 2 (1/4) — (1/4) 1 — (1/3) (1) (1/5) — F-adapt 9/51 — 4/513/54/5— 1/31/3 unif 4/54/56/52/5 ‘An optimal design example’ R-adapt 1 4/56/52/5 F-adapt 1 4/51 2/5 Convergence rates are given as powers of the representative mesh-size 1/ ndof which is proportional to H on uniform grids. Unavailable values are left blank, non-continuous rates are put in parantheses, inconclusive convergence behaviour is marked by “—”. Boiger and Carstensen Advanced Modeling and Simulation in Engineering Sciences 2013, 1:5 Page 17 of 23 http://www.amses-journal.com/content/1/1/5 Figure 4 Convergence plot of the modified two-well benchmark. Errors and error estimators of the modified two-well benchmark of Subsection ‘Modified two-well benchmark’, plotted against the number of degrees of freedom. Section ‘Conclusions’. Beyond those general conclusions, this example demonstrates the difficulties with ill-conditioned Hessians. While the unstabilised method reaches 10 degrees of freedom without difficulty on uniform meshes, the adapted algorithms fail without stabilisation beyond 687 324 degrees of freedom (η -adaptive) and 33 169 degrees of freedom (η -adaptive). MATLAB’s error message “Input to EIG must not contain NaN or Inf” indicates that a matrix operation returned non-finite numbers let fminunc break down. Undisplayed numerical experiments show condition numbers up to 10 and Figure 5 Initial grid for the three-well benchmark. Initial grid for the three-well benchmark of Subsection ‘Three-well benchmark’. Boiger and Carstensen Advanced Modeling and Simulation in Engineering Sciences 2013, 1:5 Page 18 of 23 http://www.amses-journal.com/content/1/1/5 Figure 6 Convergence plot of the three-well benchmark. Errors and error estimators of the three-well benchmark of Subsection ‘Three-well benchmark’, plotted against the number of degrees of freedom. beyond. The empirical convergence rates for uniform and R- as well as F-adapted mesh- refining are collected in Table 1. Moreover, Figure 1 in Section ‘Background’ reveals that stabilisation not only remedies ill-conditioned Hessians but thereby indeed allows for reduced errors in the discrete solution. An optimal design example The energy density of the topology optimisation problem of [3,8,29-33] reads W (F) := φ(|F|) for F ∈ R ⎪ t for 0  t  λ, √ √ √ √ with φ(t) := λ/2 + 2 λ(t − λ/2) for λ  t  2 λ, ⎪ √ t /2 + λ for t  2 λ. This leads to problem (2.3) with β = 0, λ = 0.0084, u ≡ 0and f ≡ 1. Since regu- larity of the solutions is unclear, only the results of Sections ‘Global convergence’, ‘A posteriori error estimates’, ‘Refined analysis for an interface model problem’ and ‘Numerical experiments’ apply. As initial triangulation T ,weuse thecoarsestcross triangulation T ={conv{(0, 0), (1, 0), (0, 1)},conv{(1, 0), (0, 1), (1, 1)}} of  = (0, 1) . The algorithms of Subsection ‘Numerical algorithms’ ran with and without stabilisa- tion for uniform and adaptive meshes and led to Figure 7 with the overall observations of Section ‘Conclusions’. The empirical convergence rates for uniform and R- as well as F-adapted mesh-refining are collected in Table 1. Undocumented experiments with a modified lower-order term f and known exact solution u led to the same convergence rates of the error estimators and confirm their accuracy. Boiger and Carstensen Advanced Modeling and Simulation in Engineering Sciences 2013, 1:5 Page 19 of 23 http://www.amses-journal.com/content/1/1/5 Figure 7 Convergence plot of the optimal design benchmark. Error estimators of the optimal design example of Subsection ‘An optimal design example’, plotted against the number of degrees of freedom. Discussion of Empirical Convergence Rates Global convergence without regularity assumptions Theorem 3.1 asserts that σ − σ  , βu − u  2 ,and |u | all tend to zero L () L () as H → 0. The plain convergence result applies to all examples from Subsections ‘Two-well benchmark’, ‘An optimal design example’, ‘Three-well benchmark’, and ‘An optimal design example’ for the uniform mesh-refinements with H = H /2. The numerical experiments, however, show empirical convergence rates displayed in the first columns of Table 1. The adaptive algorithms do not reflect the condition H → 0 explicitly and hence convergence is not guaranteed a priori. Undisplayed inves- tigations show that indeed in the R-adapted version of the three-well example of Subsection ‘Three-well benchmark’, this condition H → 0 does not appear to be true for more than 4 978 degrees of freedom. In all other experiments we observe convergence rates even for unstabilised discretisations. Empirical convergence rates for interface model problems Theorem 5.1 provides an a priori error estimate and an estimate of the stabil- isation norm. It applies to the benchmark of Subsections ‘Two-well benchmark’, ‘An optimal design example’, ‘Three-well benchmark’, and ‘An optimal design example’ only, because of β> 0 and Example 3.3, and the smoothness conditions imposed upon u from Section ‘Refined analysis for an interface model problem’. Recall the definitions of T ( ),  and  from Section ‘Refined analysis for an interface model problem’ −1 and assume u ≈ 1 ≈u , |T ( )|≈ H and | |≈ H in this dis- 2 C 2,p  , L (\ ) W ( ) 2/p cussion. This leads to a convergence rate of H for the right-hand side of Theorem 5.1. The observed convergence rates of σ − σ and u − u  2 for the stabilised L () L () Boiger and Carstensen Advanced Modeling and Simulation in Engineering Sciences 2013, 1:5 Page 20 of 23 http://www.amses-journal.com/content/1/1/5 benchmark examples in Table 1 show convergence rates beyond those guaranteed in Theorem 5.1. Theorem 5.3 implies, up to perturbations on the boundary, 1/3 −1/2 1/2 1/2 D(u − u )  u − u  +|u | + H |u | u − u  . 2     2 L () L () L () Since the exact solutions of Subsections ‘Two-well benchmark’, ‘An optimal design example’, ‘Three-well benchmark’, and ‘An optimal design example’ are all smooth up to a one-dimensional interface line, Theorem 5.3 applies to these examples. The experiments shows that the right-hand side of Theorem 5.3 is dominated by −1/2 1/2 1/2 H |u | u − u  in all examples and that the inequality is satisfied. L () Reliability without regularity assumptions Up to boundary terms, Theorem 4.1 states 2 2 σ − σ  + βu − u   η u − u  . 1,p 2 F p W () L () L () The convergence rates confirm this assertion for the general and rough estimate u − u  1,p  1 in the sense that the rates for η are worse than or equal to those W () 2 2 of σ − σ  and u − u  . In the numerical examples, u − u  1 is com- 2 H () L () L () puted and displayed in Table 1 and the convergence rates of the product u − u  1 η H () 2 2 can be compared with those of σ − σ  +u − u  . This comparison con- L () L () firms the above a posteriori error estimate. In the examples with p = 2 (of Subsections ‘Two-well benchmark’, ‘An optimal design example’, ‘Three-well benchmark’, and ‘An optimal design example’), there holds even equality of the convergence rates which demonstrates the efficiency of the estimate of Theorem 4.1. Efficiency without regularity assumptions Up to oscillations and the (possibly) higher-order term (id − I )σ , Theorem 4.2 F, q L () states η  σ − σ  + βu − u  . F  p  p L () L () The displayed convergence rates of Table 1 confirm this estimate. Reliability of the refined a posteriori error control Theorem 5.4 applies to the example of Subsection ‘Two-well benchmark’ and states 2 2 2 σ − σ  +u − u   η and D(u − u )  η . 2 L  2 H L () L () L () Table 1 confirms this estimate and shows that the estimators η and η accu- L H rately predict the convergence rate of the errors, even with equality of the conver- gence rates in the case of adaptive mesh refinements in the examples of Subsections ‘Two-well benchmark’, ‘An optimal design example’, ‘Three-well benchmark’, and ‘An optimal design example’. All displayed convergence rates of η arebetteroratleast equaltothose of η . For instance, for uniform mesh-refining in Subsections ‘Two-well benchmark’, ‘An optimal design example’, ‘Three-well benchmark’, and ‘An optimal design example’, 2 2 the error terms σ − σ  +u − u  converge with the empirical convergence L () L () rate 7/5 while the upper bound η does so with a reduced convergence rate 4/5. The F Boiger and Carstensen Advanced Modeling and Simulation in Engineering Sciences 2013, 1:5 Page 21 of 23 http://www.amses-journal.com/content/1/1/5 refined error estimator η is a guaranteed upper bound (via Theorem 5.4) and converges with an empirical convergence rate 1. Performance of the minimisation algorithm 1 In all numerical experiments of this paper, Algorithm 1 reaches the maximal number 3 of iterations. While this suggests that the optimal s is not found after three iterations, undisplayed experiments with higher iteration counts and hence higher computational efforts result solely in marginal improvements. Conclusions Effects of stabilisation The empirical convergence rates of the error estimators η , η and the errors F R u − u  2 and σ − σ  for uniform mesh-refinement with and without sta- L () L () bilisation coincide. This indicates that the choice γ = 1 leads to some significant perturbation but maintains the correct convergence rate at the same time. This is differ- ent for adaptive mesh refinement with less optimal convergence rates. Our conclusion is that an improved adaptive algorithm has to be developed with balance of local mesh- refinement and global stabilisation parameters in future research. The tested algorithm from Subsection ‘Numerical algorithms’ does neither reflect the effects of stabilisation nor that of inexact solve. Another important aspect of the stabilisation is the regularisation of the Hessian in the step SOLVE of Subsection ‘Numerical algorithms’. In the three-well problem of Subsection ‘Three-well benchmark’, the unstabilised adaptive algorithms fail. Adaptive versus uniform mesh-refinement The overall empirical convergence rates of the errors and estimators of the unstabilised computation for adaptive mesh-refinements are better than those for uniform mesh- refinements. This is in contrast to the stabilised computation, where the true errors σ − σ  and u − u  2 behave better for uniform compared with the two adap- L () L () tive mesh-refinments (with the exception in Subsection ‘An optimal design example’ where there is equality). It is observed that adaptivity does not necessarily improve the converegnce rates of the error σ − σ and u − u  2 in a stabilised compu- L () L () tation. Surprisingly, the convergence of the gradient errors D(u − u ) 2 are slightly L () improved in the instabilised calculation by adaptive mesh-refinements. The adaptive mesh-refinement is expected to reduce the a posteriori error estimators in the first place: cf. [1,34] for the estimator reduction property. Indeed, the convergence rates of the a posteriori error estimators η , η , η , η are improved (or optimal) for adap- R F L H tive mesh-refinements (except for the three-well example of Subsection ‘Three-well benchmark’). Strong convergence of the gradients The convergence of the gradient error of the stabilised problem surpasses the expecta- tions of [12] in Subsection ‘An optimal design example’ but fails to do so in Subsections ‘Two-well benchmark’, ‘An optimal design example’, ‘Three-well benchmark’, and ‘An optimal design example’. The improved error estimator η shows the same convergence rate as the error of the gradients in Subsections ‘Two-well benchmark’, Boiger and Carstensen Advanced Modeling and Simulation in Engineering Sciences 2013, 1:5 Page 22 of 23 http://www.amses-journal.com/content/1/1/5 ‘An optimal design example’, ‘Three-well benchmark’, and ‘An optimal design example’. This holds for uniform and for adapted mesh refinements and suggests that η is in fact reliable and efficient for β> 0. Guaranteed error control The assertion on η in Theorem 4.1 is reflected in the numerical examples in that the stress approximations converge faster than η in all cases. This suggests that the esti- mate u − u   1 is by far too pessimistic. In fact, the benchmark examples 1,p W () with known exact solution fulfil σ − σ   η u − u  . Similar affirmative 2 F H () L () conclusions follow for Theorem 4.2 and 5.4. Reliability-efficiency gap In comparison with the residual-based error estimator of [7,18], the new a posteriori error estimators η and η of Theorem 5.4 lead to refined error control. The improve- L H ment is marginal for uniform meshes without stabilisation but significant for adaptive stabilised computations. η and η match the convergence of the errors and so narrow L H the reliability-efficiency gap. 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Studies in Mathematics and its Applications, vol. 8. North-Holland Publishing Co., Amsterdam. p 776 34. Cascón JM, Kreuzer C, Nochetto RH, Siebert KG (2008) Quasi-optimal convergence rate for an adaptive finite element method. SIAM J Numer Anal 46(5): 2524–2550 doi:10.1186/2213-7467-1-5 Cite this article as: Boiger and Carstensen: A posteriori error analysis of stabilised FEM for degenerate convex minimisation problems under weak regularity assumptions. Advanced Modeling and Simulation in Engineering Sciences 2013 1:5 Submit your manuscript to a journal and benefi t from: 7 Convenient online submission 7 Rigorous peer review 7 Immediate publication on acceptance 7 Open access: articles freely available online 7 High visibility within the fi eld 7 Retaining the copyright to your article Submit your next manuscript at 7 springeropen.com

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