High-Order AFEM for the Laplace–Beltrami Operator: Convergence Rates
Adaptive Finite Element methods
A posteriori error estimates
Fecha de publicación
A. Bonito, J.M. Cascón, K. Mekchay, P. Morin, R.H. Nochetto. (2016) High-Order AFEM for the Laplace-Beltrami Operator: Convergence Rates. Foundations of Computational Mathematics. Vol 16 (6), pp 1473-1539
Serie / N.º
Foundations of Computational Mathematics;Volume 16, Issue 6
We present a new AFEM for the Laplace–Beltrami operator with arbitrary polynomial degree on parametric surfaces, which are globally W1∞ and piecewise in a suitable Besov class embedded in C1,α with α∈(0,1]. The idea is to have the surface sufficiently well resolved in W∞1 relative to the current resolution of the PDE in H1. This gives rise to a conditional contraction property of the PDE module. We present a suitable approximation class and discuss its relation to Besov regularity of the surface, solution, and forcing. We prove optimal convergence rates for AFEM which are dictated by the worst decay rate of the surface error in W∞1 and PDE error in H1.