An augmented galerkin procedure for the boundary integral method applied to two-dimensional screen and crack problems

Here we present a new solution procedure for Helm-holtz and Laplacian Dirichlet screen and crack problems in IR2 via boundary integral equations of the first kind having as an unknown the jump of the normal derivative across the screen or a crack curve T. Under the assumption of local finite energy we show the equivalence of the integral equations and the original boundary value problem. Via the method of local Mellin transform in [5]-[lo] and the calculus of pseudodifferential operators we derive existence, uniqueness and regularity results for the solution of our boundary integral equations together with its explicit behaviour near the screen or crack tips.With our integral equations we set up a Galerkin scheme on T and obtain high quasi-optimal convergence rates by using special singular elements besides regular splines as test and trial functions.