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

An adaptive stabilized trace finite element method for surface PDEs