High Order Cut Finite Elements for the Elastic Wave Equation
This work provides a stable high-order numerical method for elastic wave propagation on unfitted meshes, benefiting computational geophysics and engineering simulations with complex geometries.
The paper develops a high order cut finite element method for the elastic wave equation that handles boundaries/interfaces cutting through the background mesh, using stabilizing terms to ensure stability regardless of cut size. Numerical experiments in 2D verify optimal convergence rates and stability.
A high order cut finite element method is formulated for solving the elastic wave equation. Both a single domain problem and an interface problem are treated. The boundary or interface are allowed to cut through the background mesh. To avoid problems with small cuts, stabilizing terms are added to the bilinear forms corresponding to the mass and stiffness matrix. The stabilizing terms penalize jumps in normal derivatives over the faces of the elements cut by the boundary/interface. This ensures a stable discretization independently of how the boundary/interface cuts the mesh. Nitsche's method is used to enforce boundary and interface conditions, resulting in symmetric bilinear forms. As a result of the symmetry, an energy estimate can be made and optimal order a priori error estimates are derived for the single domain problem. Finally, numerical experiments in two dimensions are presented that verify the order of accuracy and stability with respect to small cuts.