A stable cut finite element method for partial differential equations on surfaces: The Helmholtz-Beltrami operator

We consider solving the surface Helmholtz equation on a smooth two dimensional surface embedded into a three dimensional space meshed with tetrahedra. The mesh does not respect the surface and thus the surface cuts through the elements. We consider a Galerkin method based on using the restrictions of continuous piecewise linears defined on the tetrahedra to the surface as trial and test functions. Using a stabilized method combining Galerkin least squares stabilization and a penalty on the gradient jumps we obtain stability of the discrete formulation under the condition h kappa < C, where h denotes the mesh size, kappa the wave number and C a constant depending mainly on the surface curvature kappa, but not on the surface/mesh intersection. Optimal error estimates in the H-1 and L-2-norms follow. (C) 2019 Elsevier B.V. All rights reserved.