Interior Estimates of Finite Volume Element Methods Over Quadrilateral Meshes for Elliptic Equations.

In this paper, we study the interior error estimates of a class of finite volume element methods (FVEMs) over quadrilateral meshes for elliptic equations. We first derive the global H-1 -and L-2-norms error estimates for a general case that the exact solution might be singular, namely, u is an element of H3/2+epsilon with epsilon > 0 arbitrarily small. These estimates generalize the existing results that were established under the regularity assumption u is an element of H-2. Then, we establish negative-norm error estimates for solutions with different regularity conditions. Finally, we study the interior estimates to show that the interior error of the FVEMs is bounded by the combination of the best local approximation error and a proper negative-norm error. We provide numerical results to verify our interior estimates.