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

In this paper,we study the coercivity of threeQ₁-finite volume element schemes for anisotropic diffusion problem.The three schemes are obtained from theQ₁-finite volume element method based on the trapezoidal rule,midpoint rule and Simpson's rule,respectively.For theQ₁-finite volume element scheme based on the trapezoidal rule,two methods are used to study the coercivity.In the first way,the element stiffness matrix is contracted to a 3×3 matrix.A necessary and sufficient condition for the positive definite of the 3×3 matrix is obtained,and then a sufficient condition for ensuring the coercivity of the scheme is obtained.Secondly,according to the bilinear element functional of the scheme,the element stiffness matrix is transformed into a new 3×3 element matrix.The new element stiffness matrix is studied as a whole,and a necessary and sufficient condition for its positive definiteness is obtained.Based on this sufficient and necessary condition,a sufficient condition of the coercivity is obtained.It is worth mentioning that the sufficient conditions obtained by the two methods are the same.This condition has an explicit form.For any meshes and any diffusion tensor,we can directly judge whether this sufficient condition is satisfied.When the diffusion tensor is the identity matrix,the sufficient condition includes the parallelogram meshes,some trapezoidal meshes and some kite meshes.For theQ₁-finite volume element scheme based on the midpoint rule,the two methods to obtain the coercivity are similar to those of theQ₁-finite volume element scheme based on the trapezoidal rule.The sufficient conditions obtained by the two methods are the same.The sufficient condition also has an explicit form,and also covers the parallelogram meshes,some trapezoidal meshes and some kite meshes.For theQ₁-finite volume element scheme based on the Simpson's rule,the second method is used to analyze the coercivity.Similarly,a sufficient condition for the coercivity of the scheme is obtained,and the condition is explicit.It is easy to judge whether the coercivity holds for any diffusion tensor and any meshes.Finally,some numerical examples are presented to validate the theoretical analysis.