AFEM for the Laplace-Beltrami Operator on Graphs: Design and Conditional Contraction Property

Abstract

We present an adaptive finite element method (AFEM) of any polynomial degree for the Laplace-Beltrami operator on $C^1$ graphs $\Gamma$ in $\R^d$, $(d\ge2)$. We first derive residual-type a posteriori error estimates that account for the interaction of both the energy error in $H^1(\Gamma)$ and the surface error in $W^1_\infty(\Gamma)$ due to approximation of $\Gamma$. We devise a marking strategy to reduce the total error estimator, namely a suitably scaled sum of the energy, geometric, and inconsistency error estimators. We prove a conditional contraction property for the sum of the energy error and the total estimator; the conditional statement encodes resolution of $\Gamma$ in $W^1_\infty$. We conclude with one numerical experiment that illustrates the theory.