A structure preserving finite element scheme for the incompressible GPR model

The Godunov–Peshkov–Romenski (GPR) model provides a unified continuum-mechanics framework for the description of both solids and fluids. In this presentation, I introduce a novel finite element discretization for the incompressible GPR model that preserves the underlying geometric structure of the equations. The proposed method is pressure-robust and conserves the curl of the distortion field in the solid regime. Time discretization is achieved through a splitting strategy combined with the average vector field method, yielding exact energy preservation or dissipation at the discrete level. I further discuss possible upwind and stabilization techniques and analyze their interaction with the structure-preserving properties of the scheme. While many ideas can be adapted from incompressible magnetohydrodynamics, which exhibits a closely related mathematical structure, the incompressible GPR model introduces additional challenges, such as the non-quadratic nature of its energy functional. The presentation concludes with numerical experiments conducted using the open-source finite element library NGSolve. This is a joint work with Michael Dumbser (University of Trento).