New Higher-Order Mass-Lumped Tetrahedral Elements for Wave Propagation Modelling

TL;DR

This paper introduces new higher-order mass-lumped tetrahedral elements with fewer nodes, improving efficiency and maintaining optimal convergence for wave propagation modeling.

Contribution

It proposes a less restrictive accuracy condition enabling construction of new tetrahedral elements of degrees 2 to 4 with fewer nodes, enhancing efficiency.

Findings

New elements require fewer nodes per element.

The method converges with optimal order in $L^2$ and energy norms.

Numerical tests confirm improved efficiency and accuracy.

Abstract

We present a new accuracy condition for the construction of continuous mass-lumped elements. This condition is less restrictive than the one currently used and enabled us to construct new mass-lumped tetrahedral elements of degrees 2 to 4. The new degree-2 and degree-3 tetrahedral elements require 15 and 32 nodes per element, respectively, while currently, these elements require 23 and 50 nodes, respectively. The new degree-4 elements require 60, 61 or 65 nodes per element. Tetrahedral elements of this degree had not been found yet. We prove that our accuracy condition results in a mass-lumped finite element method that converges with optimal order in the $L^2$-norm and energy-norm. A dispersion analysis and several numerical tests confirm that our elements maintain the optimal order of accuracy and show that the new mass-lumped tetrahedral elements are more efficient than the current ones.