Some Research On Finite Volume Element Method For Second Order Elliptic Equation

For the two-dimensional convection-diffusion convection-dominated equations,the stability,convergence and the maximum principle of the upwind finite volume element method(UFVEM)on a rectangular mesh are studied.The trail function space is the bilinear finite element space corresponding to the rectangular grid,and the test space is denoted by a piecewise constant function space on a standard center-dual partition.In the definition of the UFVEM,the upwind technique is applied to deal with the convection term.Firstly,the stability and H~1-norm error estimation of the upwind finite volume element method are proved.Secondly,if the rectangular mesh satisfies a certain ratio,then the maximum principle and the maximum norm error estimate could be obtained.In the end,some numerical experiments are presented to demonstrate the validity of the method for convection-dominated models.We propose a new Hermite cubic finite volume element scheme for Poisson equation which have an optimal convergence rate in L~2-norm.The standard Hermite cubic finite element space on triangular partition is chosen as the trail function space.There are two types of functions in the test function space,which are the piecewise linear function on the dual element surrounding the vertex of the triangular element and the piecewise constant function on the dual element surrounding the barycenter point.The difficulty of its L~2error estimation is that the approximation ability of the piecewise constant test function is weaker than the piecewise linear test function.This difficulty led to the L~2error estimation of the Hermite cubic finite volume format,which was constructed very early,has not been proven.To this end,we construct a new dual partition,so that an orthogonal condition is satisfied on the dual element around the barycenter point,and the optimal L~2error estimation is completed with the help of this orthogonal condition.A constrained finite volume element method which satisfying the discrete maximum principle is developed for the anisotropic diffusion equation with reaction terms on arbi-trary quadrilateral and triangular grids.Based on Algebra Flux Correction(AFC)method,the entries of the stiffness matrix formed by symmetric finite volume element method are split into two part,the diffusive part and the anti-diffusive part.By introducing a suit-able limiter,the anti-diffusive part does not generate new extreme values,and a nonlinear finite volume element scheme that maintains the principle of discrete extreme values is finally obtained.Numerical examples show that the constrained finite volume element method achieves the same convergence rate as the non-constrained scheme for anisotropic diffusion problems with smooth solutions on a twisted grid.And,the scheme guarantees the discrete maximum principle on a twisted grid.