Local inverse estimates for non-local boundary integral operators

Abstract

Abstract. We prove local inverse-type estimates for the four non-local boundary integral operators associated with the Laplace operator on a bounded Lipschitz domain Ω in R d for d ≥ 2 with piecewise smooth boundary. For piecewise polynomial ansatz spaces and d ∈ {2, 3}, the inverse estimates are explicit in both the local mesh width and the approximation order. An application to efficiency estimates in a posteriori error estimation in boundary element methods is given.

Congruences of convex algebras

“…By proving certain (local) inverse-type estimates [1], we are able to conclude that ρ ℓ satisfies the estimator reduction estimate…”

“…For an overview, we refer to the recent PhD thesis [10], where also the hypersingular integral equation is analyzed. Finally, we also refer to [1], where the present analysis is used to show convergence of an adaptive FEM-BEM coupling method by means of the estimator reduction principle.…”

“…We now specify the four modules of algorithm (1). For solve , we use a Galerkin BEM to approximate the solution φ of (2) by a piecewise constant QUASI-OPTIMAL ADAPTIVE BEM 3 function Φ ℓ ∈ P 0 (T ℓ ) which is given as the solution of the linear system…”

“…Second, if it is not, to find out where to refine the mesh to obtain the optimal improvement in accuracy. The latter leads immediately to the idea of the following adaptive feedback loop solve −→ estimate −→ mark −→ refine (1) where the computed solution is checked for accuracy and then improved by marking elements with big error contributions and by refining at least these elements to obtain an improved mesh. Hereafter, the index ℓ indicates the iteration index in the adaptive loop of (1).…”

Adaptive BEM for elliptic PDE systems, part I: abstract framework, for weakly-singular integral equations

“…In particular, optimal convergence of adaptive mesh-refining algorithms has been proved for polyhedral boundaries [1][2][3] as well as smooth boundaries [4]. The work [5] allows to transfer these results to piecewise smooth boundaries; see also the discussion in the review article [6]. In [7], these results have been generalized to the Helmholtz problem.…”