Cut Finite Element Methods for Elliptic Problems on Multipatch Parametric Surfaces
This work provides a numerical method for PDEs on complex surfaces with trimmed patches, relevant for computational geometry and engineering applications.
The authors develop a cut finite element method for solving elliptic problems on multipatch parametric surfaces, using Nitsche's method for interface continuity and stabilization for cut elements. Numerical examples confirm theoretical error estimates in energy and L2 norms.
We develop a finite element method for the Laplace--Beltrami operator on a surface described by a set of patchwise parametrizations. The patches provide a partition of the surface and each patch is the image by a diffeomorphism of a subdomain of the unit square which is bounded by a number of smooth trim curves. A patchwise tensor product mesh is constructed by using a structured mesh in the reference domain. Since the patches are trimmed we obtain cut elements in the vicinity of the interfaces. We discretize the Laplace--Beltrami operator using a cut finite element method that utilizes Nitsche's method to enforce continuity at the interfaces and a consistent stabilization term to handle the cut elements. Several quantities in the method are conveniently computed in the reference domain where the mappings impose a Riemannian metric. We derive a priori estimates in the energy and $L^2$ norm and also present several numerical examples confirming our theoretical results.