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We present an adaptive finite element method (AFEM) of any polynomial degree for the Laplace-Beltrami operator on C 1 graphs in R d (d 2). We first derive residual-type a posteriori error estimates that account for the interaction of both the energy error in H 1 () and the surface error in W 1 1() due to approximation of . 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 in W 1 1. We conclude with one numerical experiment that illustrates the theory.

AFEM for the Laplace-Beltrami operator on graphs: Design and conditional contraction property