The outstanding problem of finding a simple Muskhelishvili-type equation for the stress problem on multiply connected 2D domains is solved: we derive a Fredholm second kind integral equation with non-singular operators and where the unknown quantity is the interior limit of an analytic function. In a numerical application the stress field is resolved to a relative precision of 10^-10 on a large, yet simply reproducible, setup containing 4096 holes and cracks. About 50 GMRES iterations are needed for full convergence. Comparison with previous results in the literature indicates that general purpose FEM software may perform better than many special purpose codes based on classic integral equations.

The present work is a part in a greater effort which aims at constructing fast and stable integral equation based software for fracture mechanics problems. Unlike most actors in this field we choose to base our algorithms solely on Fredholm equations of the second kind. This gives our algorithms superior stability. Standard practice in computational Solid Mechanics is work with Fredholm first kind equations. While such equations are simpler to derive than second kind equations they are more difficult to solve as they typically lead to system matrices whose condition number grow with the number of discretization points and as they often need numerical preconditioning -- not unlike FEM.

I hope the talk will result in a discussion!