Pressure robust mixed methods for nearly incompressible elasticity

TL;DR

This paper extends pressure robust discretization techniques from incompressible fluids to nearly incompressible elasticity, enabling accurate solutions with standard finite elements without discontinuous methods.

Contribution

It demonstrates that gradient robustness methods for Stokes flow can be adapted to nearly incompressible elastic materials without using discontinuous finite elements.

Findings

Pressure robust methods reduce pressure influence in elasticity simulations.

Reconstruction operators effectively produce divergence-free functions.

The approach simplifies implementation compared to discontinuous methods.

Abstract

Within the last years pressure robust methods for the discretization of incompressible fluids have been developed. These methods allow the use of standard finite elements for the solution of the problem while simultaneously removing a spurious pressure influence in the approximation error of the velocity of the fluid, or the displacement of an incompressible solid. To this end, reconstruction operators are utilized mapping discretely divergence free functions to divergence free functions. This work shows that the modifications proposed for Stokes equation by Linke (2014) also yield gradient robust methods for nearly incompressible elastic materials without the need to resort to discontinuous finite elements methods as proposed in Fu, Lehrenfeld, Linke, Streckenbach (2021).