Geodesic
The gradient superconvergence of the finite volume method for a nonlinear elliptic problem of nonmonotone type
Applications of Mathematics, Tome 60 (2015) no. 5, pp. 573-596

Voir la notice de l'article provenant de la source Czech Digital Mathematics Library

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 $\|u-u_h\|_1$.
DOI : 10.1007/s10492-015-0112-8
Classification : 65M15, 65M60
Keywords: finite volume method; nonlinear elliptic problem; local and global superconvergence in the $W^{1, \infty }$-norm; a posteriori error estimator 
@article{10_1007_s10492_015_0112_8,
     author = {Zhang, Tie and Zhang, Shuhua},
     title = {The gradient superconvergence of the finite volume method for a nonlinear elliptic problem of nonmonotone type},
     journal = {Applications of Mathematics},
     pages = {573--596},
     publisher = {mathdoc},
     volume = {60},
     number = {5},
     year = {2015},
     doi = {10.1007/s10492-015-0112-8},
     mrnumber = {3396481},
     zbl = {06486926},
     language = {en},
     url = {http://geodesic.mathdoc.fr/articles/10.1007/s10492-015-0112-8/}
}
TY  - JOUR
AU  - Zhang, Tie
AU  - Zhang, Shuhua
TI  - The gradient superconvergence of the finite volume method for a nonlinear elliptic problem of nonmonotone type
JO  - Applications of Mathematics
PY  - 2015
SP  - 573
EP  - 596
VL  - 60
IS  - 5
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/articles/10.1007/s10492-015-0112-8/
DO  - 10.1007/s10492-015-0112-8
LA  - en
ID  - 10_1007_s10492_015_0112_8
ER  - 
%0 Journal Article
%A Zhang, Tie
%A Zhang, Shuhua
%T The gradient superconvergence of the finite volume method for a nonlinear elliptic problem of nonmonotone type
%J Applications of Mathematics
%D 2015
%P 573-596
%V 60
%N 5
%I mathdoc
%U http://geodesic.mathdoc.fr/articles/10.1007/s10492-015-0112-8/
%R 10.1007/s10492-015-0112-8
%G en
%F 10_1007_s10492_015_0112_8
Zhang, Tie; Zhang, Shuhua. The gradient superconvergence of the finite volume method for a nonlinear elliptic problem of nonmonotone type. Applications of Mathematics, Tome 60 (2015) no. 5, pp. 573-596. doi: 10.1007/s10492-015-0112-8

Cité par Sources :