[Paper Review] A stabilized trace finite element method for partial differential equations on evolving surfaces

Christoph Lehrenfeld, Maxim A. Olshanskii+1 more|arXiv (Cornell University)|Sep 21, 2017

Advanced Numerical Methods in Computational Mathematics22 references4 citations

TL;DR

This paper presents a stabilized trace finite element method (TraceFEM) for solving partial differential equations on evolving surfaces using a fixed background mesh and time-discrete finite differences. The method ensures optimal convergence and robust conditioning by stabilizing the trace formulation and extending solutions to a volumetric neighborhood, achieving optimal error estimates and condition number bounds independent of surface position.

ABSTRACT

In this paper, we study a numerical method for the solution of partial differential equations on evolving surfaces. The numerical method is built on the stabilized trace finite element method (TraceFEM) for the spatial discretization and finite differences for the time discretization. The TraceFEM uses a stationary background mesh, which can be chosen independent of time and the position of the surface. The stabilization ensures well-conditioning of the algebraic systems and defines a regular extension of the solution from the surface to its volumetric neighborhood. Having such an extension is essential for the numerical method to be well-defined. The paper proves numerical stability and optimal order error estimates for the case of simplicial background meshes and finite element spaces of order $m\ge1$. For the algebraic condition numbers of the resulting systems we prove estimates, which are independent of the position of the interface. The method allows that the surface and its evolution are given implicitly with the help of an indicator function. Results of numerical experiments for a set of 2D evolving surfaces are provided.

Motivation & Objective

• To develop a robust, unfitted finite element method for solving PDEs on time-dependent surfaces without requiring surface reconstruction or space-time integration.
• To ensure numerical stability and optimal convergence rates for high-order finite elements on evolving surfaces.
• To provide rigorous error estimates and condition number bounds independent of the surface's position or geometry.
• To enable the use of level-set methods for implicit surface representation while maintaining accuracy and stability.

Proposed method

• The method employs a stabilized trace finite element formulation on a stationary background mesh, with solution traces restricted to the evolving surface at each time step.
• Stabilization terms are added to control conditioning and define a volumetric extension of the solution from the surface into its neighborhood.
• Time derivatives are approximated using standard finite differences, avoiding the need for space-time meshing or reconstruction of the space-time manifold.
• The method uses a level-set function to implicitly describe the evolving surface, enabling geometric flexibility without explicit surface tracking.
• A key component is the use of a narrow-band extension of the solution to the background domain, ensuring well-posedness and stability.
• The analysis relies on geometric mappings and trace inequalities to bound errors in the extended domain and derive optimal convergence rates.

Experimental results

Research questions

• RQ1 Can a stable and optimally convergent finite element method be developed for PDEs on evolving surfaces without requiring surface reconstruction or space-time meshing?
• RQ2 How can stabilization be designed to ensure well-conditioned algebraic systems independent of surface position?
• RQ3 What error estimates can be derived for high-order finite elements in a trace finite element framework on evolving surfaces?
• RQ4 Can the condition number of the resulting linear systems be bounded independently of the surface's location or curvature?
• RQ5 How does the method perform numerically when the surface is defined implicitly via a level-set function?

Key findings

• The method achieves optimal order error estimates for finite element spaces of order $ m \geq 1 $ on simplicial background meshes.
• The condition number of the algebraic systems is bounded independently of the surface's position or geometry, ensuring robust conditioning.
• The method allows for implicit surface description via a level-set function without requiring explicit surface tracking or reconstruction of the space-time manifold.
• Numerical experiments confirm optimal convergence rates for a set of 2D evolving surfaces, validating the theoretical findings.
• The analysis establishes that the normal derivative term in the error bound is controlled via stabilization, with the final error estimate depending on $ \delta_n $, $ h $, and $ \Delta t $.
• The method is proven stable and convergent under standard assumptions, with the final error bound scaling optimally with mesh size and time step.