High-Order AFEM for the Laplace–Beltrami Operator: Convergence Rates

Abstract

We present a new AFEM for the Laplace-Beltrami operator with arbitrary polynomial degree on parametric surfaces, which are globally W 1 ∞ and piecewise in a suitable Besov class embedded in C 1,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 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 ∞ and PDE error in H 1 .1. Introduction. Let γ be a d dimensional surface in R d+1 (d ≥ 1) either with or without boundary, which is globally Lipschitz and piecewise in a suitable Besov class embedded in C 1,α with α ∈ (0, 1]. We design and study a quasi-optimal adaptive finite element method (AFEM) to approximate the solution of( 1.1) where f ∈ L 2 (γ) and −∆ γ is the Laplace-Beltrami operator (or surface Laplacian) on γ. In addition, we impose that u = 0 on ∂γ or require that γ u = 0 if ∂γ = ∅ (with γ f = 0 for compatibility). To represent ∆ γ , one needs to describe γ mathematically using, for example, parametric representations on charts, level sets, distance functions, graphs of functions, etc. Moreover, one usually obtains approximate solutions (finite element solutions) by solving the problem on approximate polyhedral surfaces rather than the surface γ itself. Exploiting the variational structure of the Laplace-Beltrami operator, [20] gives an a priori error analysis whereas [17,16,25,6] provide a posteriori counterparts. Our present objective is to continue our research on AFEM for elliptic PDEs on surfaces initiated in [25] for graphs and extended in [6] to parametric surfaces, the latter with polynomial degree n = 1. We design herein an AFEM for parametric surfaces using C 0 finite elements of degree n ≥ 1, prove optimal convergence rates and workload estimates, and study suitable approximation classes for the triple (u, f, γ).High-order finite elements are superior to linears for geometric problems: they provide better approximation of important geometric quantities such as curvature,

Approximating solutions of linear elliptic PDE's on a smooth manifold using local kernel

“…While the implementation detail of each of the existing methods is different in its own way, most of them have a unifying theme: they require a representation of the surface to approximate the tangential derivatives along the surface. For example, the finite element method (FEM) uses (triangulated) meshes to approximate the surface [5,6,7]. The approach in [8,9] represents the surface using level sets.…”

A Posteriori Error Estimates for the Laplace--Beltrami Operator on Parametric $C^2$ Surfaces

“…In [12] an estimator equivalent to λ is used in order to control the geometric consistency error. However, λ has a heuristic a priori order of h k for smooth surfaces, and the corresponding a priori error estimate is (1.3).…”

“…These different levels of precision with respect to constants reflect three different situations: In the next subsection we are concerned about verifying assumptions which assure the validity of our estimates and thus are as precise as possible concerning constants, including global constants such as the Lipschitz constant L. Global constants such as L are typically hidden as we do in when proving residual-type estimates, but it is often desirable to retain relevant local geometric information such as curvature variation in our estimates so that this information is taken into account when driving refinement in adaptive FEM. Finally, it becomes highly technical to reflect local geometric information when proving convergence of AFEM as in [12], so it is sometimes convenient to hide such information as we do in .…”

“…An alternate approach to representing γ in the context of a posteriori error estimation and adaptivity is given in [12,13]. In these works γ is only assumed to be globally Lipschitz and elementwise C 1,α .…”

“…In the latter case, adaptive algorithms based on G P generally resolve the geometry much more than is necessary to reach a given error tolerance. Quasi-optimal error decay for adaptive finite element approximations of u lying in certain regularity classes is derived in [12,13]. However, these regularity classes are artificially restricted by the surface approximation when γ is C 2 .…”