Adaptive 2D IGA boundary element methods

Abstract

We derive and discuss a posteriori error estimators for Galerkin and collocation IGA boundary element methods for weakly-singular integral equations of the first-kind in 2D. While recent own work considered the Faermann residual error estimator for Galerkin IGA boundary element methods, the present work focuses more on collocation and weightedresidual error estimators, which provide reliable upper bounds for the energy error. Our analysis allows piecewise smooth parametrizations of the boundary, local mesh-refinement, and related standard piecewise polynomials as well as NURBS. We formulate an adaptive algorithm which steers the local mesh-refinement and the multiplicity of the knots. Numerical experiments show that the proposed adaptive strategy leads to optimal convergence, and related IGA boundary element methods are superior to standard boundary element methods with piecewise polynomials.

Optimal convergence for adaptive IGA boundary element methods for weakly-singular integral equations

“…However, a detailed proof is given only for . For this inequality can be shown exactly as in the proof of [24, Lemma 4.5], where only is considered. This, together with (2.8), implies for any connected with thatThe hidden constant in (4.8) depends only on and .…”

“…We aim to apply Lemma 6.1 and show in the following that (A1)–(A2) hold for all even with and that (A3) holds for all . Then, Lemma 6.1 shows convergence (3.10) of the Faermann estimator.Of Lemma 6.1 follows immediately from [24, Theorem 4.3], where the constant depends only on , and .Can be proved exactly as in [23, Section 2.4] as is efficient (see [25, Theorem 3.1]) and has a semi-norm structure. The constant depends only on .…”

Adaptive isogeometric boundary element methods with local smoothness control

“…(b) Algorithm 3.1 allows the choice ϑ = 0 and M − ℓ = ∅, and then formally coincides with the adaptive algorithm from [FGHP16] for the weakly-singular integral equation.…”

“…Rate-optimal adaptive strategies for IGAFEM have been proposed and analyzed independently in [BG17,GHP17] for IGAFEM, while the earlier work [BG16] proves only linear convergence. As far as IGABEM is concerned, available results focus on the weakly-singular integral equation with energy space H −1/2 (Γ); see [FGP15,FGHP16] for a posteriori error estimation as well as [FGHP17] for the analysis of a rate-optimal adaptive IGABEM in 2D, and [Gan17] for corresponding results for IGABEM in 3D with hierarchical splines. Recently, [FGPS18] investigated optimal preconditioning for IGABEM in 2D with locally refined meshes.…”

An adaptive IgA‐BEM with hierarchical B‐splines based on quasi‐interpolation quadrature schemes

“…While adaptive BEM has been widely studied in the literature, see, eg, the work of Feischl et al for a recent review, the theory of adaptivity for isogeometric BEMs is still at a preliminary stage. A posteriori error analysis and refinement algorithms in the 2D setting have been presented in the works of Feischl et al In an another work of Feischl et al, the optimal convergence of adaptive IgA‐BEM for weakly singular equations was also proven. In all these studies, the adaptive scheme relies on the locally refinable nature of classical univariate B‐splines, the standard spline basis adopted in CAD and IgA.…”