Stability and Superconvergence of a Special $Q_1$-Finite Volume Element Scheme over Quadrilateral Meshes

Abstract

This paper studied the stability and superconvergence of a special isoparametric bilinear finite volume element scheme for anisotropic diffusion problems over quadrilateral meshes, where the scheme is obtained by employing the edge midpoint rule to approximate the line integrals in classical $Q_1$-finite volume element method. It can be checked that the scheme is identical to the standard five-point difference scheme for a special case. By element analysis approach, we suggest a sufficient condition to guarantee the stability of the scheme. This condition has an analytic expression, which covers the traditional $h^{1+\gamma}$-parallelogram and some trapezoidal meshes with any full anisotropic diffusion tensor. Moreover, based on the $h^2$-uniform quadrilateral mesh assumption, we proved the superconvergence $|u_I−u_h|_1=\mathcal{O}(h^2),$ where $u_I$ is the isoparametric bilinear interpolation of exact solution $u,$ and $u_h$ is the numerical solution. As a byproduct, we obtained the optimal $H^1$ and $L^2$ error estimates. Finally, the theoretical results are verified by some numerical experiments.