Condition number of the BEM matrix arising from the Stokes equations in 2D

We study the condition number of the system matrices that appear in the boundary element method when solving the Stokes equations at a 2D domain. At the boundary of the domain we impose Dirichlet conditions or mixed conditions. We show that for certain critical boundary contours the underlying boundary integral equation is not uniquely solvable. As a consequence, the condition number of the system matrix of the discrete equations is infinitely large. Hence, for these critical contours the Stokes cannot be solved by the boundary element method. To overcome this problem the domain can be rescaled. Several numerical examples are provided to illustrate the solvability problems at the critical contours.