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 the stability and the error estimate, we propose the minimum V-angle condition on tetrahedral meshes, and the surface and volume orthogonal conditions on dual meshes. Through the technique of element analysis, the local stability is equivalent to a positive definiteness of a 9 × 9 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, and 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 H1 and L2 error estimates, respectively, where the orthogonal conditions play an important role in ensuring the optimal L2 convergence rate. Numerical experiments are presented to illustrate our theoretical results.