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Superconvergence analysis and extrapolation of quasi-Wilson nonconforming finite element method for nonlinear Sobolev equations

Superconvergence analysis and extrapolation of quasi-Wilson nonconforming finite element method... Quasi-Wilson nonconforming finite element approximation for a class of nonlinear Sobolev equations is discussed on rectangular meshes. We first prove that this element has two special characters by novel approaches. One is that (∇ h (u − I h u),∇ h v h ) h may be estimated as order O(h 2) when u ∈ H 3(Ω), where I h u denotes the bilinear interpolation of u, v h is a polynomial belongs to quasi-Wilson finite element space and ∇ h denotes the piecewise defined gradient operator, h is the mesh size tending to zero. The other is that the consistency error of this element is of order O(h 2)/O(h 3) in broken H 1-norm, which is one/two order higher than its interpolation error when u ∈ H 3(Ω)/H 4(Ω). Then we derive the optimal order error estimate and superclose property via mean-value method and the known high accuracy result of bilinear element. Furthermore, we deduce the global superconvergence through interpolation post processing technique. At last, an extrapolation result of order O(h 3), two order higher than traditional error estimate, is obtained by constructing a new suitable extrapolation scheme. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Acta Mathematicae Applicatae Sinica Springer Journals

Superconvergence analysis and extrapolation of quasi-Wilson nonconforming finite element method for nonlinear Sobolev equations

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References (41)

Publisher
Springer Journals
Copyright
Copyright © 2013 by Institute of Applied Mathematics, Academy of Mathematics and System Sciences, Chinese Academy of Sciences and Springer-Verlag Berlin Heidelberg
Subject
Mathematics; Applications of Mathematics; Math Applications in Computer Science; Theoretical, Mathematical and Computational Physics
ISSN
0168-9673
eISSN
1618-3932
DOI
10.1007/s10255-013-0216-4
Publisher site
See Article on Publisher Site

Abstract

Quasi-Wilson nonconforming finite element approximation for a class of nonlinear Sobolev equations is discussed on rectangular meshes. We first prove that this element has two special characters by novel approaches. One is that (∇ h (u − I h u),∇ h v h ) h may be estimated as order O(h 2) when u ∈ H 3(Ω), where I h u denotes the bilinear interpolation of u, v h is a polynomial belongs to quasi-Wilson finite element space and ∇ h denotes the piecewise defined gradient operator, h is the mesh size tending to zero. The other is that the consistency error of this element is of order O(h 2)/O(h 3) in broken H 1-norm, which is one/two order higher than its interpolation error when u ∈ H 3(Ω)/H 4(Ω). Then we derive the optimal order error estimate and superclose property via mean-value method and the known high accuracy result of bilinear element. Furthermore, we deduce the global superconvergence through interpolation post processing technique. At last, an extrapolation result of order O(h 3), two order higher than traditional error estimate, is obtained by constructing a new suitable extrapolation scheme.

Journal

Acta Mathematicae Applicatae Sinica – Springer Journals

Published: Apr 10, 2013

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