A boundary element method for the Dirichlet eigenvalue problem of the Laplace operator

The solution of eigenvalue problems for partial differential operators byusing boundary integral equation methods usually involves some Newton potentialswhich may be resolved by using a multiple reciprocity approach. Here we proposean alternative approach which is in some sense equivalent to the above. Instead of alinear eigenvalue problem for the partial differential operator we consider a nonlineareigenvalue problem for an associated boundary integral operator. This nonlineareigenvalue problem can be solved by using some appropriate iterative scheme, herewe will consider a Newton scheme.We will discuss the convergence and the boundaryelement discretization of this algorithm, and give some numerical results.