Robust finite element discretizations for a simplified Galbrun's equation

TL;DR

This paper investigates stable finite element discretizations for a simplified, indefinite vector PDE related to Galbrun's equation, establishing conditions for robustness and stability through a Helmholtz-type decomposition.

Contribution

It introduces conditions for stable finite element schemes for a simplified Galbrun's equation, linking stability to well-understood problems like Stokes and elasticity.

Findings

Helmholtz-type decomposition is key for stability

Stable discretizations are connected to Stokes and elasticity problems

Conditions for robustness are identified for finite element schemes

Abstract

Driven by the challenging task of finding robust discretization methods for Galbrun's equation, we investigate conditions for stability and different aspects of robustness for different finite element schemes on a simplified version of the equations. The considered PDE is a second order indefinite vector-PDE which remains if only the highest order terms of Galbrun's equation are taken into account. A key property for stability is a Helmholtz-type decomposition which results in a strong connection between stable discretizations for Galbrun's equation and Stokes and nearly incompressible linear elasticity problems.