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.

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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: $\max \nolimits _{P\in S}|(\nabla u-\overline {\nabla }u_h)(P)|=O(h^2)\mathopen |\ln h|^{{3}/{2}}$, 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 
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     title = {The gradient superconvergence of the finite volume method for a nonlinear elliptic problem of nonmonotone type},
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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. http://geodesic.mathdoc.fr/articles/10.1007/s10492-015-0112-8/

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