Convergence of adaptive boundary element methods

Adaptive boundary element methods for optimal convergence of point errors

One particular strength of the boundary element method is that it allows for a high-order pointwise approximation of the solution of the related partial differential equation via the representation formula. However, the high-order convergence and hence accuracy usually suffers from singularities of the Cauchy data. We propose two adaptive mesh-refining algorithms and prove their quasi-optimal c...

Convergence of Adaptive Finite Element Methods

Title of dissertation: CONVERGENCE OF ADAPTIVE FINITE ELEMENT METHODS Khamron Mekchay, Doctor of Philosophy, 2005 Dissertation directed by: Professor Ricardo H. Nochetto Department of Mathematics We develop adaptive finite element methods (AFEMs) for elliptic problems, and prove their convergence, based on ideas introduced by Dörfler [7], and Morin, Nochetto, and Siebert [15, 16]. We first stud...

Adaptive Finite Element Methods with convergence rates

Quasi-optimal Convergence Rate for an Adaptive Boundary Element Method

For the simple layer potential V that is associated with the 3D Laplacian, we consider the weakly singular integral equation V φ = f . This equation is discretized by the lowest order Galerkin boundary element method. We prove convergence of an h-adaptive algorithm that is driven by a weighted residual error estimator. Moreover, we identify the approximation class for which the adaptive algorit...

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

In a recent work (Feischl et al. in Eng Anal Bound Elem 62:141-153, 2016), we analyzed a weighted-residual error estimator for isogeometric boundary element methods in 2D and proposed an adaptive algorithm which steers the local mesh-refinement of the underlying partition as well as the multiplicity of the knots. In the present work, we give a mathematical proof that this algorithm leads to con...