Construction and analysis of the quadratic finite volume methods on tetrahedral meshes

Abstract

A family of quadratic finite volume method (FVM) schemes are constructed and analyzed over tetrahedral meshes. In order to prove stability and error estimate, we propose the minimum V-angle condition on tetrahedral meshes, and the surface and volume orthogonal conditions on dual meshes. Through the element analysis technique, the local stability is equivalent to a positive definiteness of a $9\times9$ element matrix, which is difficult to analyze directly or even numerically. With the help of the surface orthogonal condition and congruent transformation, this element matrix is reduced into a block diagonal matrix, then we carry out the stability result under the minimum V-angle condition. It is worth mentioning that the minimum V-angle condition of the tetrahedral case is very different from a simple extension of the minimum angle condition for triangular meshes, while it is also convenient to use in practice. Based on the stability, we prove the optimal $ H^{1} $ and $L^2$ error estimates respectively, where the orthogonal conditions play an important role in ensuring optimal $L^2$ convergence rate. Numerical experiments are presented to illustrate our theoretical results.