A rigorous justification and estimation of the rate of convergence for the partial domain method in two-dimensional eigenvalue problems for the Laplace operator

The partial domain method for problems of the type indicated in the title of this paper is treated as a version either of Weinstein's method of intermediate problems or of the Ritz method. This yields a rigorous justification of the method and enables one to estimate its rate of convergence. The justification technique is demonstrated in detail for the problem of the normal modes of an $\mathrm L$-shaped membrane clamped at its edges.