Adaptive BEM with optimal convergence rates for the Helmholtz equation

Abstract

We analyze an adaptive boundary element method for the weakly-singular and hypersingular integral equations for the 2D and 3D Helmholtz problem. The proposed adaptive algorithm is steered by a residual error estimator and does not rely on any a priori information that the underlying meshes are sufficiently fine. We prove convergence of the error estimator with optimal algebraic rates, independently of the (coarse) initial mesh. As a technical contribution, we prove certain local inverse-type estimates for the boundary integral operators associated with the Helmholtz equation.

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

“…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. In recent years, we have also shown optimal convergence of adaptive isogeometric BEM (IGABEM) using one-dimensional splines for the 2D Laplace problem [8,9].…”

An efficient preconditioner for adaptive Fast Multipole accelerated Boundary Element Methods to model time-harmonic 3D wave propagation

“…The influence of the mesh discretization on the accuracy of a numerical solution still poses a challenge to the BEM community [9,5,20]. Anisotropic features of a solution (e.g., some elastic materials) as well as discontinuities near geometric singularities (e.g., corners and edges) are difficult to capture and ultimately diminish the regularity of the boundary solution and subsequent performance of a BEM.…”

“…To this end, iterative mesh refinement schemes have been proposed to transform an initial mesh into an improved one according to error estimates calculated at each step, with the goal of reducing the number of degrees of freedom required to resolve a solution within a desired level of accuracy. Most work to date for BEMs has been confined to isotropic techniques based on a posteriori error analysis from which error indicators can be derived, see e.g., [5,9] for wave propagation problems. These indicators steer refinement by systematically marking and subdividing only elements where the error is above a specified threshold-a process known as Dörfler marking.…”

“…Not only is this process unable to produce anisotropic discretizations, it is also heavily reliant on discretization strategy and model [28] (e.g., often requiring the more computationally expensive Galerkin approach). Finally, this strategy is not optimal for high-order discretizations [9].…”

“…If an extensive literature exists for volume-based methods [1,34], relatively little research attention has been paid to error estimators and mesh adaptation techniques in the BEM community. In general, most adaptation techniques for BEMs have centered on isotropic techniques [29,5,9,44] that improve performance but do not always recover optimal convergence rates for 3D problems with anisotropic features [4]. In this work, we employ a recently introduced anisotropic mesh adaptation (AMA) strategy using a metric-based error estimator whose effectiveness in recovering convergence order was first demonstrated for volumetric (finite element) methods [51,52] and only recently for first-order BEMs [20].…”