A Stable Cut Finite Element Method for Partial Differential Equations on Surfaces: The Helmholtz-Beltrami Operator
This work provides a numerical method for solving surface PDEs on complex geometries without conforming meshes, which is important for applications in computational geometry and physics.
The paper presents a stable cut finite element method for solving the Helmholtz-Beltrami equation on surfaces embedded in 3D, using a mesh that does not conform to the surface. The method achieves stability under the condition hk < C and yields optimal error estimates in H1 and L2 norms.
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 $h k < C$, where $h$ denotes the mesh size, $k$ the wave number and $C$ a constant depending mainly on the surface curvature $κ$, but not on the surface/mesh intersection. Optimal error estimates in the $H^1$ and $L^2$-norms follow.