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.

翻译：本文提出了一种自适应版本的稳定迹有限元方法（TraceFEM），旨在利用嵌入空间中的形状规则体网格求解水平集曲面上的低正则性椭圆问题。基于多项式族$Q_1$和$Q_2$，针对一阶和二阶连续迹空间，考虑了两种稳定化变体：梯度跳跃面稳定化和法向梯度体积稳定化。我们提出了一种实用的误差指示器，用于估计背景网格面上有限元解导数的"跳跃"，该指示器避免了沿离散曲面单元隐式定义的曲线边缘进行任何量的积分。对于体单元上的$Q_1$族分段三线性多项式，结合所提出的误差指示器，求解-估计-标记-细化策略实现了典型二维问题的最优收敛率。我们还提供了后验误差估计，建立了$Q_1$和$Q_2$单元以及两种稳定化类型的误差指示器的可靠性。在数值实验中，我们评估了误差指示器的可靠性和效率。尽管发现两种稳定化方法性能相当，但最低阶有限元空间似乎是自适应TraceFEM框架中更鲁棒的选择。