[Paper Review] A stabilized cut finite element method for partial differential equations on surfaces: The Laplace-Beltrami operator

Erik Burman, Peter Hansbo+1 more|arXiv (Cornell University)|Dec 4, 2013

Advanced Numerical Methods in Computational Mathematics9 references4 citations

TL;DR

This paper proposes a stabilized cut finite element method for solving the Laplace-Beltrami equation on surfaces embedded in a tetrahedral mesh, using continuous piecewise linear functions restricted to the surface. A consistent stabilization term controlling normal gradient jumps ensures optimal condition number bounds independent of surface location and enables optimal $O(h^2)$ convergence in energy and $L^2$ norms.

ABSTRACT

We consider solving the Laplace-Beltrami problem on a smooth two dimensional surface embedded into a three dimensional space meshed with tetrahedra. The mesh does not respect the surface and thus the surface cuts through the elements. We consider a Galerkin method based on using the restrictions of continuous piecewise linears defined on the tetrahedra to the surface as trial and test functions. The resulting discrete method may be severely ill-conditioned, and the main purpose of this paper is to suggest a remedy for this problem based on adding a consistent stabilization term to the original bilinear form. We show optimal estimates for the condition number of the stabilized method independent of the location of the surface. We also prove optimal a priori error estimates for the stabilized method.

Motivation & Objective

• Address the severe ill-conditioning of standard cut finite element methods when solving PDEs on surfaces with unfitted meshes.
• Overcome the sensitivity of condition number to surface location relative to the background mesh.
• Develop a consistent stabilization term that maintains optimal convergence rates while improving numerical stability.
• Establish theoretical guarantees for condition number and error estimates independent of surface geometry.
• Provide a robust framework for complex surface PDEs, including future extensions to Helmholtz and coupled bulk-surface problems.

Proposed method

• Use continuous piecewise linear finite elements defined on background tetrahedral elements and restrict them to the surface.
• Introduce a stabilization term that penalizes jumps in the normal gradient across element faces intersecting the surface.
• Formulate a consistent Galerkin method with a stabilized bilinear form to improve conditioning.
• Leverage discrete Poincaré-type estimates to bound the condition number independently of surface location.
• Apply lifting operators to map discrete functions on the surface to the continuous surface for error analysis.
• Use tangential calculus and surface differential operators (Laplace-Beltrami) to define the weak formulation on the surface.

Experimental results

Research questions

• RQ1 Can a consistent stabilization term be designed to eliminate the ill-conditioning of unfitted finite element methods on surfaces?
• RQ2 Does the stabilized method achieve optimal convergence rates in energy and $L^2$ norms for the Laplace-Beltrami problem?
• RQ3 Can the condition number of the discrete system be bounded independently of the surface's position relative to the background mesh?
• RQ4 How does the stabilization affect the conditioning and convergence behavior in practice compared to the unstabilized method?
• RQ5 Can the theoretical results be extended to more complex surface PDEs, such as the Helmholtz equation?

Key findings

• The stabilized method achieves an optimal condition number bound of $O(h^{-2})$ independent of the surface's location within the background mesh.
• Optimal $O(h^2)$ convergence rates are proven in both the energy and $L^2$ norms for the discrete solution.
• Numerical experiments confirm that the stabilized method maintains optimal convergence rates, while the unstabilized method exhibits poor conditioning and spurious zero eigenvalues.
• The condition number grows as $O(h^{-2})$ without stabilization, but stabilization significantly improves conditioning, especially when combined with diagonal scaling.
• The stabilization term effectively controls the normal derivative of discrete functions and prevents ill-conditioning due to arbitrary surface placement.
• The method remains robust even when the surface cuts through elements, and the stabilization is consistent with the weak formulation.