Some Improvements of Convergence Order of Finite Volume Solutions

In this article, we improve the convergence order of some finite volume solutions approximating some second order elliptic problems. We prove that finite volume approximations of order $O(h^{k+1})$, with $k$ integer, can be obtained after $k$ corrections, starting with finite volume solution of order $O(h)$, by using the same matrix and changing only the second member of the original system. This is done for general smooth second order elliptic problems in one dimension and for second order elliptic problems of the form $-\Delta u+pu=f$, with Dirichlet conditions. Numerical tests justifying theoretical results and showing the efficiency of the method are presented. The idea used behind these results is the one of Fox's difference correction in the context of finite difference method.