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 is 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.

In this work, we discuss and compare three methods for the numerical approximation of constant-and variable-coefficient diffusion equations in both single and composite domains with possible discontinuity in the solution/flux at interfaces, considering (i) the Cut Finite Element Method; (ii) the Difference Potentials Method; and (iii) the summationby-parts Finite Difference Method. First we give a brief introduction for each of the three methods. Next, we propose benchmark problems, and consider numerical tests-with respect to accuracy and convergence-for linear parabolic problems on a single domain, and continue with similar tests for linear parabolic problems on a composite domain (with the interface defined either explicitly or implicitly). Lastly, a comparative discussion of the methods and numerical results will be given.

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