High-Order AFEM for the Laplace-Beltrami Operator: Convergence Rates
For researchers in numerical analysis and computational geometry, this work provides a theoretical framework for high-order AFEM on surfaces, though it is incremental as it extends existing AFEM theory to a specific operator and surface class.
This paper presents a new adaptive finite element method (AFEM) for the Laplace-Beltrami operator on parametric surfaces, achieving optimal convergence rates dictated by the worst decay rate of surface and PDE errors.
We present a new AFEM for the Laplace-Beltrami operator with arbitrary polynomial degree on parametric surfaces, which are globally $W^1_\infty$ and piecewise in a suitable Besov class embedded in $C^{1,α}$ with $α\in (0,1]$. The idea is to have the surface sufficiently well resolved in $W^1_\infty$ relative to the current resolution of the PDE in $H^1$. 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 $W^1_\infty$ and PDE error in $H^1$.