The Study Of Q₁-finite Volume Element Method For Anisotropic Diffusion Problem Over Arbitrary Convex Quadrilateral Mesh

We devote the present dissertation to focus on the coercivity of the Q₁-finite volume element method?Q₁-FVEM?for anisotropic diffusion problems on arbitrary quadrilateral mesh.First,we study the coercivity of the so-called modified Q₁-finite volume scheme?mQ₁-FVEM?which is obtained by using the trapezoidal rule to ap-proximate the line integrals in the classical Q₁-FVEM.A necessary and sufficient condition is obtained for the positive definiteness of a certain element stiffness ma-trix.Based on this result,a sufficient condition is suggested to guarantee the coerciv-ity of the scheme on arbitrary convex quadrilateral meshes.The sufficient condition reduces to a geometric one,covering some standard meshes,such as the traditional h1+?-parallelogram meshes and some trapezoidal meshes.More interesting is that,this sufficient condition has explicit expression,by which one can easily judge on any diffusion tensor and any mesh with any mesh size h>0.The H¹ error estimate of the mQ₁-FVEM is obtained without the traditional h1+?-parallelogram assumption.Since the gradients of the Q₁-finite element basis functions along the bound-aries of dual cells are generally rational functions,mQ₁-FVEM and Q₁-FVEM are different.Moreover,the coercivity analysis of mQ₁-FVEM cannot be extended to Q₁-FVEM.Thus,we find a new approach to study the coercivity of Q₁-FVEM on general quadrilateral meshes.Based on the matrix expression of ???,we can transform the original cell stiffness matrix into a 3 x 3 matrix by a delicious skill.We easily obtain a necessary and sufficient condition for its positive definiteness of the new matrix.In light of this result,a sufficient condition is suggested to guarantee the coercivity of the scheme.We also find that this sufficient condition covers the traditional h1+?-parallelogram mesh assumption and has an explicit expression,by which one can easily judge on any diffusion tensor and any mesh with arbitrary mesh size h>0.Moreover,the H¹ error estimate for Q₁-FVEM is obtained without the h1+?-parallelogram assumption.We propose a new Q₁-finite volume element method?sQ₁-FVEM?for anisotrop-ic diffusion problems on convex quadrilateral meshes by using a special quadrature formula to approximate the line integrals in the classical Q₁-FVEM.We rigorously prove the coercivity of the proposed sQ₁-FVEM under the quasi-regular mesh con-dition and without the requirement that the mesh size is sufficiently small,i.e.,the sQ₁-FVEM is unconditional stable on quasi-regular quadrilateral meshes.Based on the coercivity result for the sQ₁-FVEM,we are able to provide a new way to prove the coercivity of the classical Q₁-FVEM under the regular mesh condition.As a direct consequence,the H¹ error estimates for the sQ₁-FVEM and the classical Q₁-FVEM are also derived under the regular mesh condition.In addition,a counterex-ample is constructed to show that more restrictive mesh conditions such as the h1+?-parallelogram condition is still needed in order to obtain the L² error estimate for the sQ₁-FVEM.Finally,the application of the mQ₁-FVEM is considered.Based on a completely new approach that the mQ₁-FVEM are introduced to treat the vertex unknowns,we propose and analyze a stable nine-point scheme for diffusion problems.Due to the NPS is easy for coding,it has been widely used in some radiation hydrodynamics codes for a long time,such as LARED-I and MARED[22,69].When the new nine-point scheme is applied to such problems,we just need to add a program to achieve the values of the vertex unknowns by mQ₁-FVEM,which is easy to implement.With the aid of the theoretical results of the mQ₁-FVEM and a simple discrete functional technique[81,94],the stability result and error estimate of the resulting nine-point scheme both in H¹ norm are obtained under a standard and weak geometric assump-tion.Our proposed scheme does not possess the so-called numerical heat-barrier issue suffered by many existing cell-centered or hybrid schemes[51].