Adaptive FEM with coarse initial mesh guarantees optimal convergence rates for compactly perturbed elliptic problems

Abstract

We prove that for compactly perturbed elliptic problems, where the corresponding bilinear form satisfies a Gårding inequality, adaptive mesh-refinement is capable of overcoming the preasymptotic behavior and eventually leads to convergence with optimal algebraic rates. As an important consequence of our analysis, one does not have to deal with the a priori assumption that the underlying meshes are sufficiently fine. Hence, the overall conclusion of our results is that adaptivity has stabilizing effects and can overcome possibly pessimistic restrictions on the meshes. In particular, our analysis covers adaptive mesh-refinement for the finite element discretization of the Helmholtz equation from where our interest originated.

Adaptive BEM with optimal convergence rates for the Helmholtz equation

“…We apply Algorithm 7 to the model problem (51). Recall that the error estimator (53) satisfies Lemma 9 and model problem (51) fits in the abstract setting of [BHP17]. Verbatim argumentation as for the weakly-singular case proves Theorem 10 for the hypersingular integral equation.…”

“…Adaptive algorithm. Based on the error estimator η • we consider the following algorithm, where the expanded making strategy in Step(iv)-(v) goes back to[BHP17].Algorithm 7. Input: Parameters 0 < θ ≤ 1 and C mark ≥ 1 as well as initial triangulation T 0 with Φ −1 := 0 ∈ P p (T 0 ) and η −1 := 1.Adaptive loop: For all = 0, 1, 2, .…”

A new accurate residual-based a posteriori error indicator for the BEM in 2D-acoustics

“…In practice, all these assumptions are satisfied if the mesh T 0 corresponding to the initial space V 0 is sufficiently fine. However, [18] shows that some assumptions still hold if the initial mesh is coarse.…”

“…We do not necessarily have V ∞ = H when we use a standard autoadaptive algorithm like the so-called Dörfler marking strategy [16]. Nevertheless, the paper [18] proposes a simple modification of the latter which does not impact the optimal convergence of the algorithm and ensures the property (7).…”

Instance-Optimal Goal-Oriented Adaptivity

“…see, e.g., [BHP17,Lemma 22]. Hence, #(T \T 0 ) in (29) can, in fact, be replaced by #T (at the cost that the constant C in (4) will additionally depend on #T 0 ).…”