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 C1 graphs Γ in R{double-struck}d (d ≥ 2). We first derive residual-type a posteriori error estimates that account for the interaction of both the energy error in H1(Γ) and the surface error in W∞1 (Γ) due to approximation of Γ. 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 Γ in W∞1. We conclude with one numerical experiment that illustrates the theory.