Abstract. We analyze the simplest and most standard adaptive finite element method (AFEM), with any polynomial degree, for general second order linear, symmetric elliptic operators. As is customary in practice, the AFEM marks exclusively according to the error estimator and performs a minimal element refinement without the interior node property. We prove that the AFEM is a contraction, for the sum of the energy error and the scaled error estimator, between two consecutive adaptive loops. This geometric decay is instrumental to derive the optimal cardinality of the AFEM. We show that the AFEM yields a decay rate of the energy error plus oscillation in terms of the number of degrees of freedom as dictated by the best approximation for this combined nonlinear quantity.

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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,

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We present several applications governed by geometric PDE, and their parametric finite element discretization, which might yield singular behavior. The success of such discretization hinges on an adequate variational formulation of the Laplace-Beltrami operator, which we describe in detail for polynomial degree 1. We next present a complete a posteriori error analysis which accounts for the usual PDE error as well as the geometric error induced by interpolation of the surface. This leads to an adaptive finite element method (AFEM) and its convergence. We discuss a contraction property of AFEM and show its quasi-optimal cardinality.