# A Nitsche Method for Elliptic Problems on Composite Surfaces

## TL;DR

This paper introduces a Nitsche finite element method for solving elliptic PDEs on complex composite surfaces composed of multiple intersecting surfaces, providing stability and error estimates.

## Contribution

It develops a novel Nitsche-based finite element approach for elliptic problems on composite surfaces with interfaces, including stability analysis and numerical implementations.

## Key findings

- Method achieves stability under certain geometric assumptions
- Error estimates are derived for exact geometry representation
- Numerical examples demonstrate method effectiveness

## Abstract

We develop a finite element method for elliptic partial differential equations on so called composite surfaces that are built up out of a finite number of surfaces with boundaries that fit together nicely in the sense that the intersection between any two surfaces in the composite surface is either empty, a point, or a curve segment, called an interface curve. Note that several surfaces can intersect along the same interface curve. On the composite surface we consider a broken finite element space which consists of a continuous finite element space at each subsurface without continuity requirements across the interface curves. We derive a Nitsche type formulation in this general setting and by assuming only that a certain inverse inequality and an approximation property hold we can derive stability and error estimates in the case when the geometry is exactly represented. We discuss several different realizations, including so called cut meshes, of the method. Finally, we present numerical examples.