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

summary: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 $\|u-u_h\|_1$