Coercivity Analysis Of Some Q₁-finite Volume Element Schemes For Diffusion Problems

In this paper,we study the coercivity of two special Q₁-finite volume element schemes for anisotropic diffusion problems,including Q₁-finite volume element scheme based on the midpoint rule and the single-point rule,respectively.The first Q₁-finite volume element scheme is obtained by employing the midpoint rule to approximate the line integrals in the standard Q₁-finite volume element method.A necessary and sufficient condition for the positive definiteness of the element stiffness matrix is presented.This sufficient condition has an explicit form,containing the information of diffusion tensor and grids.It can be reduced to a pure geometric one on certain special meshes,including parallelogram meshes,H^1+y-parallelogram meshes and some trapezoidal meshes.Moreover,different from existing work,we prove rigorously the optimal H¹ error estimate without the H^1+y-parallelogram as-sumption.Numerical examples are presented to confirm our theoretical analysis.The second Q₁-finite volume element scheme is obtained by using the single-point rule to approximate the line integrals in the standard Q₁-finite volume element method.By choosing properly the location of the quadrature point,this new scheme satisfies the coercivity condition un-conditionally.Furthermore,we can obtain the optimal H¹ error estimate at the same time.Numerical examples are also provided to validate the theoretical results.