Zhongying Chen, Yuesheng Xu, Yuanyuan Zhang
Abstract: The main purpose of this paper is to study the construction of higher-order finite volume methods (FVMs) of triangle meshes. We investigate the relationship of the three theoretical notions crucial in the construction of FVMs: the uniform ellipticity of the family of its discrete bilinear forms, its inf-sup condition and its uniform local ellipticity. Both the uniform ellipticity of the family of the discrete bilinear forms and its inf-sup condition guarantee the unique solvability of the FVM equations and the optimal error estimate of the approximate solution. We characterize the uniform ellipticity in terms of the inf-sup condition and a geometric condition on the bijective operator mapping from the trial space onto the test space involved in the construction of FVMs. We present a geometric interpretation of the inf-sup condition. Moreover, since the uniform local ellipticity is a convenient sufficient condition for the uniform ellipticity, we further provide sufficient conditions and necessary conditions of the uniform local ellipticity.
Journal: Journal of Computational and Applied Mathematics
DOI: 10.1016/j.cam.2013.03.050