International Workshop on MeshFree Methods 2003 1 Numerical determination of the resonance frequencies in a bounded domain using the MFS Carlos J. S. Alves ( 1 ) and Pedro Antunes ( 2 ) Abstract: In this work we present a numerical algorithm for the determination of the eigenvalues and eigenfunctions associated to the Dirichlet problem for the Laplacian, in a bounded domain. The determination of higher eigenfrequencies is a well known numer- ical problem that has been addressed with other numerical methods. Here we propose to use the method of fundamental solutions. Since the MFS produces highly ill conditioned matrices, a particular technique was derived to overcome the difficulty of determining accurately those eigenfrequencies. Extensive numerical simulations will be presented. Keywords: eigenfrequencies, resonance, acoustic waves, method of fundamental solu- tions. 1 Introduction The determination of the resonance frequencies associated to the Laplace operator is an old mathematical problem with applications in several scientific areas (eg. [6, 7]). Several classical numerical methods for PDE’s have been used to determine both the eigenvalues and the eigenfunctions for arbitrary domains. More recently, meshfree methods using radial basis functions (eg. [5]) have been considered. Here we propose to consider an algorithm for the determination of eigenvalues and eigenfrequencies based on the method of fundamental solutions (MFS). In particular, we present several numerical experiments that show some interesting nodal domains (eg. [1]) for high Dirichlet eigenvalues associ- ated to non trivial 2D domains. The application of this type of algorithm might be made for other boundary conditions and also for the exterior problem, but for simplicity we will consider here only the 2D Dirichlet problem for bounded domains. 1 Departamento de Matem´ atica & CEMAT, Instituto Superior T´ ecnico, Av. Rovisco Pais 1, 1049-001 Lisboa, Portugal (calves @ math.ist.utl.pt). 2 Departamento de Matem´ atica, Instituto Superior T´ ecnico, Av. Rovisco Pais 1, 1049-001 Lisboa, Por- tugal (l45935 @ isabelle.math.ist.utl.pt).