A Class Of Finite Volume Method For Diffusion Equations

In the late 1970s, the integral differential value is extended to finite difference method based on triangle mesh, overcome the handles natural boundary value conditions, and improve the accuracy, It is called general finite difference method, also called the finite volume element method was put up by professor Ronghua Li. Due to its computational simplicity and preserving local conservation of certain physical quantities, has been widely used in computingThis paper mainly introduces three finite volume methods for diffusion problem proposed by Robert Eymard:A mixed finite volume scheme for anisotropic diffusion problems on any grid, Finite volume approximation of elliptical problems and convergence of an approximation approximate gradient, A cell-centred finite-volume approximation for anisotropic diffusion operators on unstructured meshes in any space dimention. They are different from the finite volume element method put up by professor Ronghua Li. The finite volume element method has two sets of subdivision, and the three formats have only one subdivision, there is no dual subdivision.And it does not establish function space, but to establish finite difference scheme the continuity of the gradient.The paper firstly introduces three different methods of theoretical part, then they were respectively of numerical experiments.For a mixed finite volume scheme for anisotropic diffusion problems on any grid, firstly the definition of admissible discretization and notations is present, then the finite volume method is proposed, this method is a mixed finite volume method, which exists two equations, we can solve the numerical solution and discrete approximation,and at the same time, in each control volume boundary we can solve the flow. This paper applied Taylor and Green formula to establish the finite volume method. In the paper we prove the convergence properties on any mesh,the order of convergence of the numerical and gradient approximation is 1/2.for the implementation, first we firstly solve the numerical solution in the boundary of each control volume, then solve the numerical solution and gradient approximation in each volume center and the flow.In Finite volume approximation of elliptical problems and convergence of an approximation approximate gradient, the definition of admissible discretization is similar to the definition in a mixed finite volume scheme for anisotropic diffusion problems on any grid, but the discretization need more requirements. The method Green formula is applied,in each control volume boundary midpoint, the discrete gradients of exact solutions on the midpoints of K and L are opposite numbers, this is why the discretization need more requirments. For gradient approximation, in order to define the discrete gradient, the paper introduces the Neumann problem. The order of convergence of the numerical and gradient approximation is 1 and the implementation is same as a mixed finite volume scheme for anisotropic diffusion problems on any grid.The discretization's requirment of a cell-centred finite-volume approximation for anisotropic diffusion operators on unstructured meshes in any space dimention is same as Finite volume approximation of elliptical problems and convergence of an approximation approximate gradient. The discrete gradient is the approximate solution of all the numerical solutions in the control volumeK and the adjacent control volume. And then this paper introduces the symmetric bilinear product.Finally using Green formula, discrete gradient and definition of the symmetric bilinear product define the finite volume method. The order of convergence of the numerical and gradient approximation is 1.This is an thirteen-point format, we can get the numerical solution and discrete gradient.