Title: Cut Finite Element Methods for Elliptic Problems on Multipatch Parametric Surfaces,
Abstract: 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.