Dörfler marking with minimal cardinality is a linear complexity problem

Abstract

Most adaptive finite element strategies employ the Dörfler marking strategy to single out certain elements M ⊆ T of a triangulation T for refinement. In the literature, different algorithms have been proposed to construct M, where usually two goals compete: On the one hand, M should contain a minimal number of elements. On the other hand, one aims for linear costs with respect to the cardinality of T . Unlike expected in the literature, we formulate and analyze an algorithm, which constructs a minimal set M at linear costs. Throughout, pseudocodes are given.i.e., the marked elements control a fixed percentage θ of the overall error estimator. Clearly, one aims to choose the set M ℓ with as few elements as possible.

The saturation assumption yields optimal convergence of two-level adaptive BEM

“…For an algorithm with linear complexity determining M satisfying (14) and (15) with C mark = 2, we refer to [Ste07]. Moreover, we refer to the recent work [PP19] for an algorithm with linear cost O(#T ) and C mark = 1.…”

An adaptive weak Galerkin finite element method with hierarchical bases for the elliptic problem

“…On the MARK step, we adopt the Dörfler marking strategy which is a mature strategy and widely used in adaptive algorithm [10]. Recently, Dörfler marking with minimal cardinality has been shown to be a linear complexity problem [25]. In this marking strategy the local error indicators { η L , j } j = 1, …, N are sorted in descending order.…”

“…Taking the difference between (13) and (25) restricted to V 0E h , and the difference between (25) and (36)…”

Goal-oriented adaptive finite element methods with optimal computational complexity

“…• the iterates u k and z k are computed in parallel and each step of the solver in Algorithm 2(i) can be done in linear complexity O(#T ), • all indicators η (T, u k ) and ζ (T, z k ) for all T ∈ T can be computed at total cost O(#T ), • the marking in Algorithm 2(iv) can be performed at linear cost O(#T ) (according to [Ste07] this can be done for the strategies outlined in Remark 1 with M having almost minimal cardinality; moreover, we refer to a recent own algorithm in [PP20] with linear cost even for M having minimal cardinality), • we have linear cost O(#T ) to generate the new mesh T +1 . Since a step ( , k) ∈ Q of Algorithm 2 depends on the full history of preceding steps, the total work spent to compute…”