Reliable and efficient a posteriori error estimation for adaptive IGA boundary element methods for weakly-singular integral equations

Abstract

We consider the Galerkin boundary element method (BEM) for weakly-singular integral equations of the first-kind in 2D. We analyze some residual-type a posteriori error estimator which provides a lower as well as an upper bound for the unknown Galerkin BEM error. The required assumptions are weak and allow for piecewise smooth parametrizations of the boundary, local mesh-refinement, and related standard piecewise polynomials as well as NURBS. In particular, our analysis gives a first contribution to adaptive BEM in the frame of isogeometric analysis (IGABEM), for which we formulate an adaptive algorithm which steers the local mesh-refinement and the multiplicity of the knots. Numerical experiments underline the theoretical findings and show that the proposed adaptive strategy leads to optimal convergence.

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

“…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 .…”

Convergence of Adaptive BEM and Adaptive FEM-BEM Coupling for Estimators Without h-Weighting Factor

“…While this is immaterial for piecewise a ne boundaries, P (T ⋆ ) depends on the chosen parametrization for non-a ne boundaries. For 2D BEM, this restriction is removed in the recent work [33].…”

Linear elastic fracture simulation directly from CAD: 2D NURBS-based implementation and role of tip enrichment

“…The IGA utilizes the same splines, that are used to exactly represent the geometry, as basis functions for the approximation of the unknown elds, which builds up a more direct link between CAD and analysis. Non-uniform rational B-splines (NURBS) based IGA has been widely investigated in many areas [50] [67]. A fast IGABEM solver has been developed in [68].…”