Optimal Preconditioners for Solving Two-dimensional Fractures and Screens Using Boundary Elements

An extensive range of problems in science and engineering involve two-dimensional crack, screen (Shestopalov, Smirnov, & Chernokozhin, 2000; Meixner, 1972; E. Stephan, 1987), or interface problems (Costabel & Dauge, 2002; Nicaise & Sändig, 1994a, 1994b; E. P. Stephan & Wendland, 1984). The simplest approach to model them is to consider the following problem for an open curve C ⊂ R, −∆U = 0 in R \ C , U = g or ∂U ∂n = h on C , (0.1) plus decay conditions at ∞, and with suitable boundary data g and h. Boundary integral methods (BEM) are an attractive option to deal with the unboundedness of the domain and the decay conditions at ∞. Unfortunately, the singular behaviour of the solutions causes the linear systems arising from the related integral operators to be numerically ill-conditioned. Therefore, iterative solvers require unreasonable computational work. This can be tackled by using suitable preconditioners (Hiptmair, 2006). This thesis presents the numerical implementation of the Calderón-type identities deduced from Jerez-Hanckes and Nédélec (Jerez-Hanckes & Nédélec, 2011; Jerez-Hanckes & Nédélec, 2012) for an open interval. In addition, they are used to build optimal preconditioners for the associated integral operators arising from (0.1) and their extension to the Helmholtz equation. Finally, since the singularities of the solutions to the associated weakly singular operator behave as 1/ √ d where d is the distance to the endpoints, we can achieve more accuracy through local refinement around the two endpoints. Therefore, this work also extends preconditioning theory to non-uniform meshes.