This paper reviews the state of the art and discusses very recent mathematical
developments in the field of adaptive boundary element methods. This includes an overview …

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 …

We consider the preconditioned conjugate gradient method (PCG) with optimal
preconditioner in the frame of the boundary element method for elliptic first-kind integral …

Abstract We introduce Neural Network (NN for short) approximation architectures for the
numerical solution of Boundary Integral Equations (BIEs for short). We exemplify the …

We consider a transmission problem consisting of a singularly perturbed reaction diffusion
equation on a bounded domain and the Laplacian in the exterior, connected through …

For the non-preconditioned Galerkin matrix of the hypersingular integral operator, the
condition number grows with the number of elements as well as the quotient of the maximal …

Functional error estimates are well-established tools for a posteriori error estimation and
related adaptive mesh-refinement for the finite element method (FEM). The present work …

We define and analyze (local) multilevel diagonal preconditioners for isogeometric
boundary elements on locally refined meshes in two dimensions. Hypersingular and weakly …

We introduce Neural Network (NN for short) approximation architectures for the numerical
solution of Boundary Integral Equations (BIEs for short). We exemplify the proposed NN …

We develop and analyze strategies to couple the discontinuous Petrov-Galerkin method with
optimal test functions to (i) least-squares boundary elements and (ii) various variants of …