A family of quadratic finite volume element schemes over triangular meshes for elliptic equations

Abstract

In this paper, we construct and analyze a family of quadratic finite volume element schemes over triangular meshes for elliptic equations. This family of schemes cover some existing quadratic schemes. For these schemes, by element analysis, we find that each element matrix can be split as two parts: the first part is the element stiffness matrix of the standard quadratic finite element method, while the second part is a tensor product of two vectors. Thanks to this finding, we obtain a sufficient condition to ensure the existence, uniqueness and coercivity result of the finite volume element solution on triangular meshes. More interesting is that, the above condition has a simple and analytic expression, and only relies on the interior angles of each triangular element. Based on this result, a minimum angle condition, better than some existing ones, can be obtained. Moreover, based on the coercivity result, we prove that the finite volume element solution converges to the exact solution with an optimal convergence rate in H1 norm. Finally, some numerical examples are provided to validate the theoretical findings.