L 2 ERROR ESTIMATES FOR A CLASS OF ANY ORDER FINITE VOLUME SCHEMES OVER QUADRILATERAL MESHES

In this paper, we propose a unified $ L^2 $ error estimate for a class of bi-$r$ finite volume (FV) schemes on a quadrilateral mesh for elliptic equations, where $r\ge 1$ is arbitrary. The main result is to show that the FV solution possesses the optimal order $L^2 $ error provided that $(u,f) \in H^{r+1} \times H^r $, where $u$ is the exact solution and $f$ is the source term of the elliptic equation. Our analysis includes two basic ideas: (1) By the Aubin--Nistche technique, the $L^2$ error estimate of an FV scheme can be reduced to the analysis of the difference of bilinear forms and right-hand sides between the FV and its corresponding finite element (FE) equations, respectively; (2) with the help of a special transfer operator from the trial to test space, the difference between the FV and FE equations can be estimated through analyzing the effect of some Gauss quadrature. Numerical experiments are given to demonstrate the proved results.