The gradient superconvergence of the finite volume method for a nonlinear elliptic problem of nonmonotone type

Tie Zhu Zhang; Shu Hua Zhang

Applications of Mathematics (2015)

  • Volume: 60, Issue: 5, page 573-596
  • ISSN: 0862-7940

Abstract

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We study the superconvergence of the finite volume method for a nonlinear elliptic problem using linear trial functions. Under the condition of C -uniform meshes, we first establish a superclose weak estimate for the bilinear form of the finite volume method. Then, we prove that on the mesh point set S , the gradient approximation possesses the superconvergence: max P ∈ S | ( ∇ u - ∇ ¯ u h ) ( P ) | = O ( h 2 ) | ln h | 3 / 2 , where ∇ ¯ denotes the average gradient on elements containing vertex P . Furthermore, by using the interpolation post-processing technique, we also derive a global superconvergence estimate in the H 1 -norm and establish an asymptotically exact a posteriori error estimator for the error ∥ u - u h ∥ 1 .

How to cite

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Zhang, Tie Zhu, and Zhang, Shu Hua. "The gradient superconvergence of the finite volume method for a nonlinear elliptic problem of nonmonotone type." Applications of Mathematics 60.5 (2015): 573-596. <http://eudml.org/doc/271591>.

@article{Zhang2015,
abstract = {We study the superconvergence of the finite volume method for a nonlinear elliptic problem using linear trial functions. Under the condition of $C$-uniform meshes, we first establish a superclose weak estimate for the bilinear form of the finite volume method. Then, we prove that on the mesh point set $S$, the gradient approximation possesses the superconvergence: $\max \nolimits _\{P\in S\}|(\nabla u-\overline\{\nabla \}u_h)(P)|=O(h^2)\mathopen |\ln h|^\{\{3\}/\{2\}\}$, where $\overline\{\nabla \}$ denotes the average gradient on elements containing vertex $P$. Furthermore, by using the interpolation post-processing technique, we also derive a global superconvergence estimate in the $H^1$-norm and establish an asymptotically exact a posteriori error estimator for the error $\Vert u-u_h\Vert _1$.},
author = {Zhang, Tie Zhu, Zhang, Shu Hua},
journal = {Applications of Mathematics},
keywords = {finite volume method; nonlinear elliptic problem; local and global superconvergence in the $W^\{1,\infty \}$-norm; a posteriori error estimator; finite volume method; nonlinear elliptic problem; local and global superconvergence in the $W^\{1,\infty \}$-norm; a posteriori error estimator},
language = {eng},
number = {5},
pages = {573-596},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {The gradient superconvergence of the finite volume method for a nonlinear elliptic problem of nonmonotone type},
url = {http://eudml.org/doc/271591},
volume = {60},
year = {2015},
}

TY - JOUR
AU - Zhang, Tie Zhu
AU - Zhang, Shu Hua
TI - The gradient superconvergence of the finite volume method for a nonlinear elliptic problem of nonmonotone type
JO - Applications of Mathematics
PY - 2015
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 60
IS - 5
SP - 573
EP - 596
AB - We study the superconvergence of the finite volume method for a nonlinear elliptic problem using linear trial functions. Under the condition of $C$-uniform meshes, we first establish a superclose weak estimate for the bilinear form of the finite volume method. Then, we prove that on the mesh point set $S$, the gradient approximation possesses the superconvergence: $\max \nolimits _{P\in S}|(\nabla u-\overline{\nabla }u_h)(P)|=O(h^2)\mathopen |\ln h|^{{3}/{2}}$, where $\overline{\nabla }$ denotes the average gradient on elements containing vertex $P$. Furthermore, by using the interpolation post-processing technique, we also derive a global superconvergence estimate in the $H^1$-norm and establish an asymptotically exact a posteriori error estimator for the error $\Vert u-u_h\Vert _1$.
LA - eng
KW - finite volume method; nonlinear elliptic problem; local and global superconvergence in the $W^{1,\infty }$-norm; a posteriori error estimator; finite volume method; nonlinear elliptic problem; local and global superconvergence in the $W^{1,\infty }$-norm; a posteriori error estimator
UR - http://eudml.org/doc/271591
ER -

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