Computational experiments in the problem on eigenvalues for the Laplace operator in the polygonal domain

Abstract

The technique of numerical evaluation of the Laplace operator eigenvalues in a polygon are described. The L-shaped domain is taken as an example. The conformal mapping of the circle is constructed to this area, using the Christoffel-Schwarz integral. In the circle, the problem is solved by the author’s (with K.I. Babenko’s contribution) procedures without saturation developed earlier. The question remains whether this procedure is applicable to piecewise-smooth boundaries (the conformal mapping has special features on the boundary). The performed computations show that it is possible to calculate about five eigenvalues (for the Neumann problem about 100 eigenvalues) of the Laplace operator in this domain with two to five characters after the decimal point.