# Cut Finite Element Methods for Elliptic Problems on Multipatch Parametric Surfaces

## TL;DR

This paper introduces a cut finite element method for solving elliptic PDEs on complex, multipatch parametric surfaces, effectively handling trimmed patches and interface continuity.

## Contribution

It develops a novel cut finite element approach with Nitsche's method for elliptic problems on multipatch surfaces, including stability and error analysis.

## Key findings

- The method achieves optimal convergence rates in energy and L2 norms.
- Numerical examples confirm theoretical stability and accuracy.
- The approach effectively manages trimmed patches and interface conditions.

## 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.