  In this paper we present an iterative method, inspired by the inverse
iteration with shift technique of finite linear algebra, designed to find the
eigenvalues and eigenfunctions of the Laplacian with homogeneous Dirichlet
boundary condition for arbitrary bounded domains $\Omega\subset R^{N}$. This
method, which has a direct functional analysis approach, does not approximate
the eigenvalues of the Laplacian as those of a finite linear operator. It is
based on the uniform convergence away from nodal surfaces and can produce a
simple and fast algorithm for computing the eigenvalues with minimal
computational requirements, instead of using the ubiquitous Rayleigh quotient
of finite linear algebra. Also, an alternative expression for the Rayleigh
quotient in the associated infinite dimensional Sobolev space which avoids the
integration of gradients is introduced and shown to be more efficient. The
method can also be used in order to produce the spectral decomposition of any
given function $u\in L^{2}(\Omega)$.
