Boundary element method for membrane and torsion crack problems

Here we present as an application of 12] an improved Galerkin method for the boundary integral equations governing a plane interface problem. Membrane and torsion crack problems can be treated by slight modifications.A tedious analysis incorporating the Mellin transform shows that the coupled system of integral equations—with some Fredholm equations of the second kind and some of the first kind—on the boundary curve Γ is strongly elliptic, i.e., there holds a Gårding inequality. This property implies convergence of almost optimal order of the Galerkin procedure. The use of singularity functions together with regular finite elements on Γ provides convergence results in a scale of Sobolev spaces and even quasi-optimal asymptotic error estimates for the stress intensity factors. These factors are computed directly by our Galerkin scheme, i.e., no additional computations are needed.The Galerkin method is implemented by an appropriate numerical integration leading to a Galerkin collocation. The latter is a modified collocation method which can be easily implemented on computing machines.