Michael Karkulik

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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(More)
We prove convergence and quasi-optimality of a lowest-order adaptive boundary element method for a weakly-singular integral equation in 2D. The adaptive meshrefinement is driven by the weighted-residual error estimator. By proving that this estimator is not only reliable, but under some regularity assumptions on the given data also efficient on locally(More)
We consider a (possibly) nonlinear interface problem in 2D and 3D, which is solved by use of various adaptive FEM-BEM coupling strategies, namely the Johnson-Nédélec coupling, the Bielak-MacCamy coupling, and Costabel’s symmetric coupling. We provide a framework to prove that the continuous as well as the discrete Galerkin solutions of these coupling(More)
We report on the Matlab program package HILBERT. It provides an easily-accessible implementation of lowest order adaptive Galerkin boundary element methods for the numerical solution of the Poisson equation in 2D. The library was designed to serve several purposes: The stable implementation of the integral operators may be used in research code. The(More)
Only very recently, Sayas (SIAM J. Numer. Anal. 2009) proved that the Johnson-Nédélec one-equation approach from (Math. Comp. 1980) provides a stable coupling of finite element method (FEM) and boundary element method (BEM). In our work, we now adapt the analytical results for different a posteriori error estimates developed for the symmetric FEM-BEM(More)
Galerkin methods for FEM and BEM based on uniform mesh refinement have a guaranteed rate of convergence. Unfortunately, this rate may be suboptimal due to singularities present in the exact solution. In numerical experiments, the optimal rate of convergence is regained when algorithms based on a-posteriori error estimation and adaptive mesh-refinement are(More)
We consider symmetric as well as non-symmetric coupling formulations of FEM and BEM in the frame of nonlinear elasticity problems. In particular, the Johnson-Nédélec coupling is analyzed. We prove that these coupling formulations are well-posed and allow for unique Galerkin solutions if standard discretizations by piecewise polynomials are employed. Unlike(More)
Newest vertex bisection (NVB) is a popular local mesh-refinement strategy for regular triangulations which consist of simplices. For the 2D case, we prove that the meshclosure step of NVB, which preserves regularity of the triangulation, is quasi-optimal and that the corresponding L-projection onto lowest-order Courant finite elements (P1-FEM) is always(More)
We analyze an adaptive boundary element method with fixed-order piecewise polynomials for the hyper-singular integral equation of the Laplace-Neumann problem in 2D and 3D which incorporates the approximation of the given Neumann data into the overall adaptive scheme. The adaptivity is driven by some residual-error estimator plus data oscillation terms. We(More)