A Coercivity Result of Quadratic Finite Volume Element Schemes over Triangular Meshes (Advances in Applied Mathematics and Mechanics, 2023)
In this work, we study the coercivity of a family of quadratic finite volume element (FVE) schemes over triangular meshes for solving elliptic boundary valueproblems. The analysis is based on the standard mapping from the trial function spaceto the test function space so that the coercivity result can be naturally incorporatedwith most existing theoretical results such as $H^1$ and $L^2$ error estimates. The noveltyof this paper is that, each element stiffness matrix of the quadratic FVE schemes can bedecomposed into three parts: the first part is the element stiffness matrix of the standard quadratic finite element method (FEM), the second part is the difference betweenthe FVE and FEM on the element boundary, while the third part can be expressedas the tensor product of two vectors. As a result, we reach a sufficient condition toguarantee the existence, uniqueness and coercivity result of the FVE solution on general triangular meshes. Moreover, based on this sufficient condition, some minimumangle conditions with simple, analytic and computable expressions are obtained. Bycomparison, the existing minimum angle conditions were obtained numerically froma computer program. Theoretical findings are conformed with the numerical results.