A Nitsche Method for Elliptic Problems on Composite Surfaces
This work provides a theoretical framework for solving PDEs on complex composite surfaces, which is relevant for computational geometry and engineering applications involving intersecting surfaces.
The paper develops a Nitsche-type finite element method for elliptic PDEs on composite surfaces, enabling the use of broken finite element spaces without continuity requirements across interfaces. Stability and error estimates are derived under general assumptions, with numerical examples demonstrating the method's applicability.
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.