A construction of higher-order finite volume methods

Chen, Zhongying; Xu, Yuesheng; Zhang, Yuanyuan

doi:10.1090/S0025-5718-2014-02881-0

A construction of higher-order finite volume methods
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by Zhongying Chen, Yuesheng Xu and Yuanyuan Zhang;

Math. Comp. 84 (2015), 599-628

DOI: https://doi.org/10.1090/S0025-5718-2014-02881-0

Published electronically: July 28, 2014

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Abstract:

We provide a method for the construction of higher-order finite volume methods (FVMs) for solving boundary value problems of the two dimensional elliptic equations. Specifically, when the trial space of the FVM is chosen to be a conforming triangle mesh finite element space, we describe a construction of the associated test space that guarantees the uniform local-ellipticity of the family of the resulting discrete bilinear forms. We show that the uniform local-ellipticity ensures that the resulting FVM has a unique solution which enjoys an optimal error estimate. We characterize the uniform local-ellipticity in terms of the uniform boundedness (below by a positive constant) of the smallest eigenvalues of the matrices associated with the FVMs. We then translate the characterization to equivalent requirements on the shapes of the triangle meshes for the trial spaces. Four convenient sufficient conditions for the family of the discrete bilinear forms to be uniformly local-elliptic are derived from the characterization. Following the general procedure, we construct four specific FVMs which satisfy the uniform local-ellipticity. Numerical results are presented to verify the theoretical results on the convergence order of the FVMs.

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