Higher-order finite volume methods for elliptic boundary value problems

Authors:
Zhongying Chen;Junfeng Wu;Yuesheng Xu
Affiliations:
Department of Scientific Computing and Computer Applications, Sun Yat-Sen University, Guangzhou, People's Republic of China 510275;Department of Scientific Computing and Computer Applications, Sun Yat-Sen University, Guangzhou, People's Republic of China 510275;Department of Scientific Computing and Computer Applications, Sun Yat-Sen University, Guangzhou, People's Republic of China 510275 and Department of Mathematics, Syracuse University, Syracuse, USA ...
Venue:
Advances in Computational Mathematics
Year:
2012

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Abstract

This paper studies higher-order finite volume methods for solving elliptic boundary value problems. We develop a general framework for construction and analysis of higher-order finite volume methods. Specifically, we establish the boundedness and uniform ellipticity of the bilinear forms for the methods, and show that they lead to an optimal error estimate of the methods. We prove that the uniform local-ellipticity of the family of the bilinear forms ensures its uniform ellipticity. We then establish necessary and sufficient conditions for the uniform local-ellipticity in terms of geometric requirements on the meshes of the domain of the differential equation, and provide a general way to investigate the mesh geometric requirements for arbitrary higher-order schemes. Several useful examples of higher-order finite volume methods are presented to illustrate the mesh geometric requirements.