Higher-order finite volume methods II: Inf–sup condition and uniform local ellipticity

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.