Stabilized finite elements for time-harmonic waves in incompressible and nearly incompressible elastic solids

The propagation of waves in elastic solids at or near the incompressible limit is of interest in many current and emerging applications. Standard low-order Galerkin finite element discretization struggles with both incompressibility and wave dispersion. Galerkin least squares stabilization is known to improve computational performance of each of these ingredients separately. A novel approach of combined pressure-curl stabilization is presented, facilitating the use of continuous, equal-order interpolation of displacements and pressure. The pressure stabilization parameter is determined by stability considerations, while the curl stabilization parameter is determined by dispersion considerations. The proposed pressure-curl-stabilized scheme provides stable and accurate results on a variety of numerical tests for incompressible and nearly incompressible elastic waves computed with linear elements.