Stability of finite volume approximations for the Laplacian operator on quadrilateral and triangular grids
type: Item
instance of: scholarly article
title: Stability of finite volume approximations for the Laplacian operator on quadrilateral and triangular grids
zbMATH Open document ID: 1065.65111
author: M. Svärd
author: Jan Nordström
published in: Applied Numerical Mathematics
publication date: 2004-10-12T00:00:00Z
review text: The objective of this paper is to analyse an edge based finite volume approximation of the Laplacian and construct an accurate and stable scheme to implement boundary conditions for time dependent problems. Of particular interest are unstructured grids where the finite volume method is fully advantageous. The heat equation is taken as a model problem. The Cauchy problem in one and several space dimensions is analyzed and the stability on unstructured grids is proven by showing the negative semidefiniteness of the matrices. Next, the initial-boundary value problem is considered and a scheme is constructed in a summation by parts framework. The boundary conditions are imposed in a stable and accurate manner using a penalty formulation. Numerical computations are performed for the wave equation in two-dimensions, verifying stability and order of accuracy for structured grids. However, the results are not satisfactory for unstructured grids, and further investigation reveals that the approximation is not consistent for general unstructured grids. However, grids consisting of equilateral polygons recover the convergence. It appears that the required stability analysis should be different from that for the heat equation.
reviewed by: Fumio Kikuchi
Mathematics Subject Classification ID: 65M06
Mathematics Subject Classification ID: 35K05
Mathematics Subject Classification ID: 35L05
Mathematics Subject Classification ID: 65M50
Mathematics Subject Classification ID: 65M12
zbMATH DE Number: 2107330
zbMATH Keywords: heat equation
zbMATH Keywords: Cauchy problem
zbMATH Keywords: initial-boundary value problem
zbMATH Keywords: wave equation
zbMATH Keywords: convergence
MaRDI profile type: Publication
full work available at URL: j.apnum.2004.02.001
OpenAlex ID: W2083302203
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DOI: 10.1016/J.APNUM.2004.02.001
description: scientific article; zbMATH DE number 2107330