Adaptive Finite Element Methods with Convergence Rates

Abstract

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EachH1/2–stable projection yields convergence and quasi–optimality of adaptive FEM with inhomogeneous Dirichlet data in Rd

“…[5,10,27], but with less dependencies for the upper bound of the adaptivity parameters. We refer to [6,15] for a characterization of approximation classes in terms of Besov regularity.…”

Convergence of an adaptive semi-Lagrangian scheme for the Vlasov-Poisson system

“…Following the rather classical idea of combining good adaption qualities and a very simple underlying structure, we will consider a particular class of adaptive triangulations that are associated to graded dyadic quadrangulations and which can also be seen as a particular case of the so called "newest vertex bisection" method described in [10] (we refer to [46] or [19] for more informations about hierarchical bases).…”

“…where the second equality comes from (10). Now because any vertex of any affine piece of f n 1 is of the form (x i , v) with x i ∈ Γ n , a little algebra shows that ρ n 1 is quadratic between two nodes of Γ n .…”

“…The adaptive discretization for achieving this prescribed accuracy has optimal complexity. A possible benchmark to properly define and evaluate this notion of optimality is provided by nonlinear approximation theory, which describes the optimal trade-off between accuracy and complexity in terms of specific BesovSobolev smoothness of the solution, see in particular [10,19,23,26]. In this paper, we propose a new adaptive semi-Lagrangian scheme, in which the first above described objective is achieved.…”

Convergence and optimal complexity of adaptive finite element eigenvalue computations

“…Although these methods have been extensively studied theoretically since Babuska and Rheinboldt [5] and have also been successful in practice and become widely spread (see, e.g., [2,6,7,10,17,[31][32][33]40,41,43,45] and the references cited therein), their convergence for boundary value problems was not shown until the work by Dörfler [19]. The result in [19] was later extended by Dörfler and Wilderotter [20], Morin et al [30,31], Veeser [39], Binev et al [9], Mekchay and Nochetto [29], Stevenson [36] and Cascon et al [14] for the standard finite element method, Carstensen and Hoppe [12] and Chen et al [16] for the mixed finite element method, and Carstensen and Hoppe [13] for the nonconfroming finite element method.…”