Computing experiments in the problem on eigenvalues for the operator of Laplace in the polygonal domain

The technique of a numerical evaluation of eigenvalues of an operator of Laplace in a polygon is described. As an example it is considered $L$-figurative area. The circle conformal mapping on this area by means of an integral of Christoffel–Schwarz is under construction. In a circle the problem dares on earlier developed by the author (together with K. I. Babenko) a technique without saturation. The problem consists in, whether this technique to piecewise smooth boundaries (the conformal mapping has on singularity boundary) is applicable. The done evaluations show that it is possible to calculate about 5 eigenvalues (for a problem of Neumann about 100 eigenvalues) an operator of Laplace in this area with two-five signs after a comma.