The Coercivity Analysis Of The Quadratic Finite Volume Element Schemes For Diffusion Problems On Triangular Meshes

In this paper,we study the coercivity of a class of quadratic finite volume element(FVEM)schemes for diffusion problems on triangular meshes.This type of schemes cover the existing quadratic schemes of Lagrangian type.Using the general mapping(?=1)from the trial function space to the test function space,we find that each element matrix can be decomposed 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 between FVEM and FEM on the cell boundary,and the third part is the tensor product of two vectors.Using this decomposition,we obtain a sufficient condition to guarantee the existence,uniqueness and coercivity of the FVEM solution on the triangular mesh.Based on this condition,we derive the minimum angle condition with an analytical expression.The second chapter of this paper introduces the main content of[12],and expounds the difference and connection between[12]and this paper.In the third chapter,we construct and analyze a class of quadratic finite volume element schemes for the isotropic steady-state diffusion problems on triangular meshes.The analysis shows that the element stiffness matrix can be decomposed into three parts.By using this decomposition,a sufficient condition is obtained to guarantee the existence,uniqueness and coercivity of the FVEM solution on the triangular mesh.In addition,a quadratic finite volume element scheme for the anisotropic diffusion problems on triangular meshes is constructed and analyzed in the fourth chapter.The analysis shows that the element stiffness matrix can also be decomposed into three parts which are easy to calculate and analyze.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,in order to validate the theoretical results in this paper,some numerical examples are given.