Published

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

Long Chen

SIAM Journal on Numerical Analysis, Volume 47, Issue 6, pp. 4021-4043

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ABSTRACT: Finite volume methods are an important class of discretization method since the conservation law is locally preserved and the capability of discretizing complex geometry domains. However it is limited by low order approximation since most finite volume methods use piecewise constant or linear function space. 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 a linear finite volume method. Our new method is modified from hierarchical basis finite element method. Optimal convergence rate in $H^1$-norm of quadratic finite volume methods for Poisson equation over two dimensional triangular and rectangular grids is obtained and numerical examples are provided to show the effective of the method.