How to avoid mass matrix for linear hyperbolic problems

TL;DR

This paper presents a method to avoid mass matrix inversion in finite element solutions of linear hyperbolic problems, simplifying computations while analyzing the impact of finite element space choices.

Contribution

It introduces a novel approach to bypass mass matrix inversion in finite element methods for hyperbolic problems, with analysis of finite element space effects.

Findings

Method successfully avoids mass matrix inversion

Numerical examples confirm correctness

Impact of finite element space choice analyzed

Abstract

We are interested in the numerical solution of linear hyperbolic problems using continuous finite elements of arbitrary order. It is well known that this kind of methods, once the weak formulation has been written, leads to a system of ordinary differential equations in $\R^N$, where $N$ is the number of degrees of freedom. The solution of the resulting ODE system involves the inversion of a sparse mass matrix that is not block diagonal. Here we show how to avoid this step, and what are the consequences of the choice of the finite element space. Numerical examples show the correctness of our approach.