An adaptive stabilized trace finite element method for surface PDEs

TL;DR

The paper develops an adaptive stabilized TraceFEM for elliptic PDEs on surfaces, using a bulk unfitted mesh and two stabilizations ($s_h^{JF}$ and $s_h^{NV}$) across low-order spaces $Q_1$ and $Q_2$. It introduces a practical a posteriori error indicator that avoids curved-edge integrations, proves its reliability, and validates it through numerical experiments on a unit sphere with low-regularity solutions. Results show that the $Q_1$ space yields robust, optimal adaptive convergence, while $Q_2$ offers comparable reliability but weaker efficiency, particularly with the NV stabilization; overall, the approach provides a scalable and implementable framework for adaptive surface FEM on unfitted grids.

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.

Figures (5)

• Figure 1: Snapshots of the mesh crosscuts at different cycles of the adaptive procedure from Section \ref{['sec:adaptiveTrace']}. The surface $\Gamma_h$ is not shown. Active elements $\mathcal{T}_h^\Gamma$ and the corresponding domain $\omega_h$ are colored by the values of the solution \ref{['polar']} with $\lambda=0.4$. Vertical direction corresponds to OZ axis. Top: the whole domain $[-2,2]^3$, with many cells remain coarse throughout the procedure. Bottom: closeup view of the north pole $(0,0,1)$ of the unit sphere where the gradient of the solution \ref{['polar']} blows up.
• Figure 2: Uniform mesh refinement for different values of $\lambda$ using the scheme \ref{['FEM']} which is based on the $Q_1$ TraceFEM and is stabilized by \ref{['s_nv']}. Left: $\|u_h-u^e\|_{L^2(\Gamma_h)}$ error. Right: $\|\nabla_{\Gamma_h}u_h-(\nabla_{\Gamma}{}u)^e\|_{L^2(\Gamma_h)}$ error. The exact solution $u_\lambda$ is of low regularity, $u\in H^{1+\lambda}(\Gamma)$ only. The expected reduction of the convergence rates to $h^{\lambda}$, for the $H^1$-seminorm is observed for $\lambda<1$. The $L^2$-norm error appears to be less sensitive to $\lambda$ at least for the tested refinement levels.
• Figure 3: Adaptive refinement with $\theta=0.5$ using the indicator \ref{['total_indicator']} for the $Q_1$ TraceFEM. Left: $s_h^{NV}$ stabilization with $\rho_S=10h^{-1}_S$. Right: $s_h^{JF}$ stabilization with $\sigma_F=10$. Top: surface errors \ref{['error_norms']} for $e_h=u_h-u^e$ and the global estimator $(\sum_T\eta^2(S_T))^{1/2}$. Bottom: efficiency indexes \ref{['def::efficiency']} for different patches of neighbors. The exact solution $u\in H^{1+\lambda}(\Gamma)$ with $\lambda=0.4$ is given by \ref{['polar']} on the unit sphere $\Gamma$. We observe that the indicator \ref{['total_indicator']} is reliable and efficient for $Q_1$ TraceFEM with both stabilizations.
• Figure 4: Adaptive refinement with $\theta=0.5$ using the indicator \ref{['total_indicator']} for the $Q_2$ TraceFEM. Left: $s_h^{NV}$ stabilization with $\rho_S=10h^{-1}_S$. Right: $s_h^{JF2}$ stabilization with $\sigma_F=\tilde{\sigma}_F=\tilde{\sigma}_\Gamma=\sigma_\Gamma=10$. Top: surface errors \ref{['error_norms']} for $e_h=u_h-u^e$ and the global estimator $(\sum_T\eta^2(S_T))^{1/2}$. The exact solution $u\in H^{1+\lambda}(\Gamma)$ with $\lambda=0.4$ is given by \ref{['polar']} on the unit sphere $\Gamma$. The indicator \ref{['total_indicator']} is reliable in the energy norm for the $Q_2$ TraceFEM with both stabilizations. The growth of all indexes shown on the bottom panels suggest the lack of efficiency. Unlike the energy norm, for which the indicator was designed for, convergence rate in $L^2$ norm appears to be suboptimal for the $s_h^{NV}$ stabilization.
• Figure 5: The effect of the stabilization parameter $\sigma_F$ on the adaptive refinement in Figure \ref{['fig:main_Q2']}. Left: $\sigma_F=0.1$. Right: $\sigma_F=1000$. Surface errors \ref{['error_norms']} for $e_h=u_h-u^e$ and the global estimator $(\sum_T\eta^2(S_T))^{1/2}$ are shown. We observe that decreasing the stabilization parameter does not improve the lack of efficiency while increasing it postpones the asymptotic regime of convergence.

Theorems & Definitions (8)

• Remark 2.1
• Lemma 3.1
• proof
• Lemma 3.2
• proof
• Lemma 3.3
• proof
• Remark 4.1