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

TL;DR

This paper develops and analyzes quadratic finite volume methods on tetrahedral meshes, introducing new mesh conditions to ensure stability and optimal error estimates, supported by theoretical proofs and numerical experiments.

Contribution

The paper introduces a novel minimum V-angle condition for tetrahedral meshes and analyzes the stability and error estimates of quadratic FVM schemes under this condition.

Findings

Proved stability under the minimum V-angle condition.

Established optimal H^1 and L^2 error estimates.

Validated theoretical results with numerical experiments.

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.