A new T-compatibility condition and its application to the discretization of the damped time-harmonic Galbrun's equation

TL;DR

This paper introduces a weaker T-compatibility condition for discretizing weakly T-coercive operators, enabling convergence proofs for finite element methods applied to the damped time-harmonic Galbrun's equation, which models stellar oscillations.

Contribution

It proposes a less restrictive criterion for operator convergence, broadening the applicability of finite element discretizations for complex wave equations.

Findings

Established a new point-wise convergence criterion for discrete operators.

Proved convergence of finite element methods for Galbrun's equation under the new framework.

Demonstrated the importance of divergence operator stability in the analysis.

Abstract

We consider the approximation of weakly T-coercive operators. The main property to ensure the convergence thereof is the regularity of the approximation (in the vocabulary of discrete approximation schemes). In a previous work the existence of discrete operators $T_n$ which converge to $T$ in a discrete norm was shown to be sufficient to obtain regularity. Although this framework proved useful for many applications for some instances the former assumption is too strong. Thus in the present article we report a weaker criterion for which the discrete operators $T_n$ only have to converge point-wise, but in addition a weak T-coercivity condition has to be satisfied on the discrete level. We apply the new framework to prove the convergence of certain $H^1$-conforming finite element discretizations of the damped time-harmonic Galbrun's equation, which is used to model the oscillations of stars. A main ingredient in the latter analysis is the uniformly stable invertibility of the divergence operator on certain spaces, which is related to the topic of divergence free elements for the Stokes equation.