A New Class of High Order Finite Volume Methods for Second Order Elliptic Equations

Abstract

In the numerical simulation of many practical problems in physics and engineering, finite volume methods are an important and popular class of discretization methods due to the local conservation and the capability of discretizing domains with complex geometry. However, they are limited by low order approximation since most existing finite volume methods use piecewise constant or linear function space to approximate the solution. In this paper, a new class of high order finite volume methods for second order elliptic equations is developed by combining high order finite element methods and linear finite volume methods. Optimal convergence rate in $H^1$-norm for our new quadratic finite volume methods over two-dimensional triangular or rectangular grids is obtained.