Optimal Biquadratic Finite Volume Element Methods on Quadrilateral Meshes

Abstract

In this paper, an optimal biquadratic finite volume element scheme is established and analyzed for elliptic equations on quadrilateral meshes. It is proved that the new scheme has optimal $O(h^2)$ convergence rate in $H^1$ norm and $O(h^3)$ convergence rate in $L^2$ norm with the assumption that each element is an $h^2$-parallelogram.