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

KHAMRON MEKCHAY; PEDRO MORIN; RICARDO H. NOCHETTO

MATHEMATICS OF COMPUTATION

American Mathematical Society

Lugar: Providence, Rhode Island, EE.UU.; Año: 2011 vol. 80 p. 625 - 648

0025-5718

We present an adaptive finite element method (AFEM) of any polynomial degree for the Laplace-Beltrami operator on $C^1$ graphs $Gamma$ in $R^d ~(dge2)$. 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.