[Paper Review] Cut Finite Element Methods for Partial Differential Equations on Embedded Manifolds of Arbitrary Codimensions

Erik Burman, Peter Hansbo+2 more|arXiv (Cornell University)|Oct 5, 2016

Numerical methods in engineering4 citations

TL;DR

This paper presents a unified theoretical framework for stabilized cut finite element methods solving the Laplace-Beltrami equation on manifolds of arbitrary codimension embedded in R^k. It introduces three novel stabilization forms—including a new surface normal gradient penalty—enabling optimal convergence rates and robust condition number bounds independent of mesh-geometry alignment, validated numerically for curves and surfaces in R^3.

ABSTRACT

We develop a theoretical framework for the analysis of stabilized cut finite element methods for the Laplace-Beltrami operator on a manifold embedded in $\\mathbb{R}^d$ of arbitrary codimension. The method is based on using continuous piecewise polynomials on a background mesh in the embedding space for approximation together with a stabilizing form that ensures that the resulting problem is stable. The discrete manifold is represented using a triangulation which does not match the background mesh and does not need to be shape-regular, which includes level set descriptions of codimension one manifolds and the non-matching embedding of independently triangulated manifolds as special cases. We identify abstract key assumptions on the stabilizing form which allow us to prove a bound on the condition number of the stiffness matrix and optimal order a priori estimates. The key assumptions are verified for three different realizations of the stabilizing form including a novel stabilization approach based on penalizing the surface normal gradient on the background mesh. Finally, we present numerical results illustrating our results for a curve and a surface embedded in $\\mathbb{R}^3$.

Motivation & Objective

• To develop a general theoretical framework for analyzing stabilized cut finite element methods on manifolds of arbitrary codimension embedded in R^k.
• To establish optimal a priori error estimates and condition number bounds for the discrete problem despite arbitrary mesh-geometry intersections.
• To verify the framework's applicability to three distinct stabilization forms, including a novel surface normal gradient stabilization.
• To extend geometric approximation estimates and interpolation error bounds to higher codimensions where standard closest point mappings are unavailable.
• To demonstrate robustness and convergence through numerical experiments on curves and surfaces in R^3 with varying mesh alignment.

Proposed method

• Uses continuous piecewise polynomial finite elements defined on a background mesh in R^k, restricted to a discrete manifold that cuts through the mesh arbitrarily.
• Introduces a stabilizing form that ensures stability and condition number control without requiring matching or shape-regular triangulations.
• Employs three stabilization mechanisms: (1) jump in normal gradient across active mesh faces, (2) full gradient on active elements, and (3) a new surface normal gradient penalty on the background mesh.
• Derives abstract conditions on the stabilization form that guarantee bounded condition number and optimal convergence rates.
• Establishes geometric approximation estimates for the discrete manifold using only approximation and regularity assumptions, avoiding explicit closest point projections.
• Applies interpolation error estimates in the general codimension setting, extending prior results beyond codimension one.

Experimental results

Research questions

• RQ1 Can a unified theoretical framework be developed for cut finite element methods on manifolds of arbitrary codimension?
• RQ2 Do the key assumptions on the stabilization form—necessary for condition number bounds and optimal error estimates—hold for multiple stabilization strategies?
• RQ3 Can a new stabilization based on the surface normal gradient on the background mesh achieve better consistency and stability than full gradient stabilization?
• RQ4 How do geometric approximation errors behave in high codimensions where closest point mappings are not explicitly available?
• RQ5 What is the dependence of the condition number on mesh-geometry alignment, and can stabilization ensure robustness across all configurations?

Key findings

• The proposed framework guarantees optimal order a priori error estimates and bounded condition number for all three stabilization forms, including the novel surface normal gradient stabilization.
• The condition number of the stiffness matrix scales as O(h^{-2}) with mesh size h, and numerical results confirm this bound across varying mesh alignments.
• The new surface normal gradient stabilization achieves significantly smaller consistency error than full gradient stabilization, enabling more flexible parameter choices and higher-order approximations.
• Numerical experiments on a surface and a curve in R^3 show robust condition numbers when stabilization is applied, especially with sufficiently large penalty parameters τ.
• The unstabilized method exhibits strong, irregular peaks in the condition number as a function of mesh-geometry alignment, while stabilized variants remain well-conditioned across all configurations.
• The mean scaled condition number h²κ(𝒜) remains bounded and converges to a constant as h→0, confirming theoretical predictions for all tested cases and stabilization forms.