# Efficient quadrature rules for computing the stiffness matrices of mass-lumped tetrahedral elements for linear wave problems

## TL;DR

This paper introduces new efficient quadrature rules for computing stiffness matrices in mass-lumped tetrahedral finite elements, enabling faster wave propagation simulations with heterogeneous materials while maintaining accuracy.

## Contribution

The paper develops novel quadrature rules for higher-degree mass-lumped tetrahedra that reduce computational cost and support heterogeneous materials without losing convergence rate.

## Key findings

- New quadrature rules for degrees 3 and 4 tetrahedra with fewer points
- Numerical results show improved efficiency over exact stiffness matrix computation
- Optimal convergence order maintained with heterogeneous materials

## Abstract

We present new and efficient quadrature rules for computing the stiffness matrices of mass-lumped tetrahedral elements for wave propagation modelling. These quadrature rules allow for a more efficient implementation of the mass-lumped finite element method and can handle materials that are heterogeneous within the element without loss of the convergence rate. The quadrature rules are designed for the specific function spaces of recently developed mass-lumped tetrahedra, which consist of standard polynomial function spaces enriched with higher-degree bubble functions. For the degree-2 mass-lumped tetrahedron, the most efficient quadrature rule seems to be an existing 14-point quadrature rule, but for tetrahedra of degrees 3 and 4, we construct new quadrature rules that require less integration points than those currently available in the literature. Several numerical examples confirm that this approach is more efficient than computing the stiffness matrix exactly and that an optimal order of convergence is maintained, even when material properties vary within the element.