An adaptive stabilized trace finite element method for surface PDEs

TL;DR

This paper presents an adaptive stabilized Trace Finite Element Method for surface PDEs that employs a novel error indicator and achieves optimal convergence rates, improving robustness and efficiency in solving elliptic problems on level-set surfaces.

Contribution

It introduces an adaptive stabilized TraceFEM with a practical error indicator, providing reliable a posteriori estimates and demonstrating optimal convergence for low-regularity surface PDEs.

Findings

The error indicator reliably estimates solution jumps and guides mesh refinement.

Both stabilization variants perform comparably, with lower degree elements showing greater robustness.

Numerical experiments confirm optimal convergence rates and efficiency of the adaptive method.

Abstract

The paper introduces an adaptive version of the stabilized Trace Finite Element Method (TraceFEM) designed to solve low-regularity elliptic problems on level-set surfaces using a shape-regular bulk mesh in the embedding space. Two stabilization variants, gradient-jump face and normal-gradient volume, are considered for continuous trace spaces of the first and second degrees, based on the polynomial families $Q_1$ and $Q_2$. We propose a practical error indicator that estimates the `jumps' of finite element solution derivatives across background mesh faces and it avoids integration of any quantities along implicitly defined curvilinear edges of the discrete surface elements. For the $Q_1$ family of piecewise trilinear polynomials on bulk cells, the solve-estimate-mark-refine strategy, combined with the suggested error indicator, achieves optimal convergence rates typical of two-dimensional problems. We also provide a posteriori error estimates, establishing the reliability of the error indicator for the $Q_1$ and $Q_2$ elements and for two types of stabilization. In numerical experiments, we assess the reliability and efficiency of the error indicator. While both stabilizations are found to deliver comparable performance,the lowest degree finite element space appears to be the more robust choice for the adaptive TraceFEM framework.