High-Order Afem For The Laplace-Beltrami Operator: Convergence Rates

BONITO, ANDREA; MORIN, PEDRO; KHAMRON MEKCHAY; CASCÓN, JOSÉ MANUEL; NOCHETTO, RICARDO H.

FOUNDATIONS OF COMPUTATIONAL MATHEMATICS

SPRINGER

Lugar: Berlin; Año: 2016 vol. 16 p. 1473 - 1539

1615-3375

We present a new AFEM for the Laplace-Beltrami operator with arbitrary polynomial degree on parametricsurfaces, which are globally $W^1_infty$ and piecewise in asuitable Besov class embedded in $C^{1,alpha}$ with $alpha in (0,1]$.The idea is to have the surfacesufficiently well resolved in $W^1_infty$ relative to the currentresolution of the PDE in $H^1$. This gives rise to a conditional contraction property of the PDE module. We present a suitable approximation classand discuss its relation to Besov regularity of the surface,solution, and forcing.We prove optimal convergence rates for AFEM which are dictatedby the worst decay rate of the surface errorin $W^1_infty$ and PDE error in $H^1$.