Error analysis of the Trefftz method for solving Laplace's eigenvalue problems

For solving Laplace's eigenvalue problems we propose new algorithms using the Trefftz method (TM) (i.e., the boundary approximation method (BAM)), by means of degeneracy of numerical Helmholtz equations. Since piecewise particular solutions can be fully adopted, the new algorithms benefit high accuracy of eigenvalues and eigenfunctions, low cost in CPU time and computer storage. Also the algorithms can be applied to solve the problems with multiple interfaces and singularities. In this paper, error estimates are derived for the approximate eigenvalues and eigenfunctions obtained. Numerical experiments for smooth and singular solutions are reported in this paper to show significance of the algorithms proposed and to verify the theoretical results made.