On adaptive wavelet boundary element methods

The present article is concerned with the numerical solution of boundary integral equa- tions by an adaptive wavelet boundary element method. This method approximates the solution with a computational complexity that is proportional to the solution’s best N -term approximation. The focus of this article is on algorithmic issues which includes the crucial building blocks and details about the efficient implementation. By numerical examples for the Laplace equation and the Helmholtz equation, solved for different geometries and right-hand sides, we validate the feasibility and efficiency of the adaptive wavelet boundary element method

An adaptive wavelet method for the solution of boundary integral equations in three dimensions

In science and engineering one often comes across partial differential equations in three dimensions, some of which can be formulated as boundary integral equations on the boundary of the three-dimensional domain of interest. With this approach the dimensionality of the problem can be reduced by one dimension and the interior as well as the exterior problem can be solved. However, this advantage does not come entirely without cost, as the involved matrices are dense. By using a wavelet scheme many matrix entries become sufficiently small such that they can be neglected without compromising the convergence rate of the underlying Galerkin scheme. In this thesis we go a step further and use an adaptive wavelet approach, meaning that specific parts of the geometry will be resolved with much detail, while other parts can stay coarse. After we have introduced the necessary theoretical foundation on boundary integral equations, wavelets and adaptive wavelet schemes, we present the details on the implementation followed by several numerical results. In the final chapter of this thesis we present the concept of goal-oriented error estimation, again followed by numerical results

On adaptive wavelet boundary element methods

The present article is concerned with the numerical solution of boundary integral equations by an adaptive wavelet boundary element method. This method approximates the solution with a computational complexity that is proportional to the solution’s best N-term approximation. The focus of this article is on algorithmic issues which includes the crucial building blocks and details about the efficient implementation. By numerical examples for the Laplace equation and the Helmholtz equation, solved for different geometries and right-hand sides, we validate the feasibility and efficiency of the adaptive wavelet boundary element method

Adaptive Wavelet BEM for boundary integral equations. Theory and numerical experiments

In this paper, we are concerned with the numerical treatment of boundary integral equations by means of the adaptive wavelet boundary element method (BEM). In particular, we consider the second kind Fredholm integral equation for the double layer potential operator on patchwise smooth manifolds contained in $\mathbb{R}^3$. The corresponding operator equations are treated by means of adaptive implementations that are in complete accordance with the underlying theory. The numerical experiments demonstrate that adaptive methods really pay off in this setting. The observed convergence rates fit together very well with the theoretical predictions that can be made on the basis of a systematic investigation of the Besov regularity of the exact solution

Adaptive Wavelet BEM for boundary integral equations. Theory and numerical experiments

We are concerned with the numerical treatment of boundary integral equations by the adaptive wavelet boundary element method. In particular, we consider the second kind Fredholm integral equation for the double layer potential operator on patchwise smooth manifolds contained in ℝ 3 . The corresponding operator equations are treated by adaptive implementations that are in complete accordance with the underlying theory. The numerical experiments demonstrate that adaptive methods really pay off in this setting. The observed convergence rates fit together very well with the theoretical predictions based on the Besov regularity of the exact solution