Analysis of a Special $Q_1$-Finite Volume Element Scheme for Anisotropic Diffusion Problems

Abstract

In this paper, we  analyze a special $Q_1$-finite volume element   scheme which is obtained by using the midpoint rule to approximate the line integrals in the standard $Q_1$-finite volume element method. A necessary and sufficient condition  for the positive definiteness of  the element stiffness matrix is  obtained. Based on  this result, a sufficient condition  for the coercivity of the scheme is  proposed. This sufficient condition has an explicit form  involving the information of the diffusion tensor and the mesh. In particular, this condition can reduce to a pure geometric one  that covers some special meshes, including the parallelogram meshes, the $h^{1+\gamma}$-parallelogram meshes and  some trapezoidal meshes. Moreover, the $H^1$ error estimate is proved rigorously without the $h^{1+\gamma}$-parallelogram assumption required by existing works. Numerical results are also presented to validate the theoretical analysis.