A Class Of Compact High Accuracy Finite Volume Schemes For Elliptic Equations With Mixed Boundary Conditions

In this paper, a class of compact finite volume schemes for one-dimensional and two-dimensional elliptic equations mixed boundary conditions are explored. The compact method is from the finite difference method,which mainly referrers to the finite difference scheme with little nodes but high precision. It is very successfully to solve the elliptic problems with first boundary value Conditions by the compact finite difference scheme,but it is difficult to deal with problems with second and third boundary conditions.As well known, the finite volume method is very convenient to deal with problems with second and third boundary conditions. In this paper,we get a class of high accuracy finite volume scheme by using finite volume method to discrete the equations and based on fully absorbing the constructing ideas of the compact difference scheme,which we call it the compact finite volume scheme.Three chapters are included. Chapter one is an introduction,in which we reviews the research progress of finite volume element method and compact difference method and outlines the main of this article. In chapter two, a compact finite volume scheme is given for mixed two point boundary value problems with both constant coefficient and variable coefficient. The linear algebraic system derived by this scheme has traditional property and can be solved by Thomas method. It is proved that the given scheme is convergent with fourth-order accuracy according to H1discrete seminorm. Furthermore, the post-processing formulas for the numerical value and derivative at midpoint of every element are obtained, which both have fourth order accuracy. Numerical examples verify the correctness of the theoretical analysis and also show the effectiveness of the scheme. In chapter three, a class of third-order accuracy finite volume scheme is given for two-dimensional elliptic equation mixed boundary conditions,by using the ideas of the finite volume discrimination method and compact difference scheme. The scheme is irreducible diagonally dominant has the properties of symmetry and positive-type,which can be solved by SOR iterative procedure. This chapter uses energy analysis method to prove the convergence. A numerical example is given to show the efficiency of the method.