PhD Thesis, Instituto Superior Tecnico, University of Lisbon, 2008

This dissertation is concerned with the numerical analysis of the Method of Fundamental Solutions (MFS), applied to Boundary Value Problems (BVP) for elliptic Partial Differential Equation (PDE) in 2D and 3D domains. In contrast to the classical element based methods, the MFS is a meshfree (and integration free) technique, where the unknown solution of the BVP is approximated by collocation on the boundary, using fundamental solutions from the underlying partial differential operator.

The applicability of the method in the case of acoustic and elastic wave propagation problems is investigated. Linear independence and density results for the corresponding fundamental solutions are proven in order to emphasize the theoretical foundations of the method. In particular, the application of the MFS to the mixed Dirichlet-Neumann BVP is justified. Also, the asymptotic behavior of the MFS is analyzed and, as a by-product, a meshfree method based on superposition of plane waves, referred to as the Plane Waves Method (PWM), is developed. Density results for the PWM are proven and the asymptotic relation between the two methods is confirmed through numerical examples.

From a numerical point of view, a motivating comparative analysis between element based and meshfree methods for the Laplace BVP is included. After illustrating the main advantages and disadvantages of the MFS, the solution of direct acoustic and elastic wave scattering problems in homogeneous media is considered. A drawback of the MFS is that it fails to provide numerical results with satisfactory accuracy for BVPs in domains with corners or cracks. This issue is circumvented by developing a hybrid method where the MFS shape functions are enriched with appropriate corner-adapted singular particular solutions of the PDE. The MFS is also shown to provide satisfactory numerical results for BVPs with noisy boundary data. Further improvement in the method's accuracy is achieved by applying regularization techniques. In the final part of this thesis, a Kansa type modication of the MFS is presented, where instead of Radial Basis Functions (RBF), Helmholtz fundamental solutions with different frequencies and source points are used as shape functions. The resulting Kansa-type MFS is applied to the solution of BVPs for second order inhomogeneous PDEs with non-constant coeficients.

CEMAT - Center for Computational and Stochastic Mathematics