Development and analysis of higher order finite volume methods over rectangles for elliptic equations
type: Item
instance of: scholarly article
title: Development and analysis of higher order finite volume methods over rectangles for elliptic equations
zbMATH Open document ID: 1020.65087
DOI: 10.1023/A:1022841012296
author: Jim jun. Douglas
author: Moongyu Park
author: Zhiqiang Cai
published in: Advances in Computational Mathematics
publication date: 2003-05-04T00:00:00Z
review text: The paper presents a systematic way to derive higher order finite volume methods (FVM) for elliptic equations. The underlying domain must allow a triangulation into rectangles. Starting point of the derivation is a mixed finite element method (FEM) and its crucial step consists in approximating a certain inner product by a quadrature rule such that the matrix which arises from applying the quadrature rule is diagonal and the numerical integration does not decrease the order of convergence of the original mixed FEM. In this way, equations in a scalar variable are obtained which define the FVM. The paper studies in detail this approach for the Brezzi-Douglas-Fortin-Marini element of index two (BDFM\(_2\)). Optimal error estimates for the scalar variable and the recovered flux are proved by considering the difference of the FVM solution to the corresponding mixed FEM solution. An analogous analysis leading to optimal error estimates is carried out for a FVM based on the Raviart-Thomas-Nedelec mixed finite element of arbitrary order in 2D and 3D. A FVM based on BDFM\(_2\) in 3D is derived and optimal error estimates are proved provided that the diffusion matrix is constant on each mesh cell. Last, it is discussed that a satisfactory FVM based on BDFM\(_3\) currently cannot be derived. Numerical examples employing the proposed FVM are not presented in this paper.
reviewed by: Volker John
Mathematics Subject Classification ID: 65N30
Mathematics Subject Classification ID: 35J25
Mathematics Subject Classification ID: 65N15
zbMATH DE Number: 1903987
zbMATH Keywords: finite volume method
zbMATH Keywords: higher order
zbMATH Keywords: rectangular element
zbMATH Keywords: mixed finite element method
zbMATH Keywords: Brezzi-Douglas-Fortin-Marini element
zbMATH Keywords: error estimates
zbMATH Keywords: Raviart-Thomas-Nedelec mixed finite element
MaRDI profile type: Publication
full work available at URL: a:1022841012296
OpenAlex ID: W2107736107
description: scientific article; zbMATH DE number 1903987