In this thesis, the boundary errors are defined for the NFM to explore the convergence rates, and the condition numbers are derived for simple cases to explore numerical stability. The optimal convergence (or exponential) rates are discovered numerically. This thesis is also devoted to seek better choice of locations for the field nodes of the FS expansions. It is found that the location of field nodes Q does not affect much on convergence rates, but do have influence on stability. Let £_ denote the distance of Q to ∂S. The larger £_ is chosen, the worse the instability of the NFM occurs. As a result, £_ = 0 (i.e., Q ∈ ∂S) is the best for stability. However, when £_ > 0, the errors are slightly smaller. Therefore, small £_ is a favorable choice for both high accuracy and good stability. This new discovery enhances the proper application of the NFM. However, even for the Dirichlet problem of Laplace¡¦s equation, when the logarithmic capacity (transfinite diameter) C_£F = 1, the solutions may not exist, or not unique if existing, to cause a singularity of the discrete algebraic equations. The problem with C_£F = 1 in the BEM is called the degenerate scale problems. The original explicit algebraic equations do not satisfy the conservative law, and may fall into the degenerate scale problem discussed in Chen et al. [15, 14, 16], Christiansen [35] and Tomlinson [42]. An analysis is explored in this thesis for the degenerate scale problem of the NFM. In this thesis, the new conservative schemes are derived, where an equation between two unknown variables must satisfy, so that one of them is removed from the unknowns, to yield the conservative schemes. The conservative schemes always bypasses the degenerate scale problem; but it causes a severe instability. To restore the good stability, the overdetermined system and truncated singular value decomposition (TSVD) are proposed. Moreover, the overdetermined system is more advantageous due to simpler algorithms and the slightly better performance in error and stability. More importantly, such numerical techniques can also be used, to deal with the degenerate scale problems of the original NFM in [15, 14, 16]. For the boundary integral equation (BIE) of the first kind, the trigonometric functions are used in Arnold [3], and error analysis is made for infinite smooth solutions, to derive the exponential convergence rates. In Cheng¡¦s Ph. Dissertation [18], for BIE of the first kind the source nodes are located outside of the solution domain, the linear combination of fundamental solutions are used, error analysis is made only for circular domains. So far it seems to exist no error analysis for the new NFM of Chen, which is one of the goal of this thesis. First, the solution of the NFM is equivalent to that of the Galerkin method involving the trapezoidal rule, and the renovated analysis can be found from the finite element theory. In this thesis, the error boundary are derived for the Dirichlet, the Neumann problems and its mixed types. For certain regularity of the solutions, the optimal convergence rates are derived under certain circumstances. Numerical experiments are carried out, to support the error made.

Le problème de l'optimisation des valeurs propres du Laplacien est ancien puisqu'à la fin du XIXème siècle Lord Rayleigh conjecturait que la première valeur propre avec condition de Dirichlet était minimisée par le disque. Depuis le problème a été beaucoup étudié. Et les possibilités de recherches sont multiples : diverses conditions, ajout de contraintes, existence, description des optima ... Dans ce document on se limite aux conditions de Dirichlet et de Neumann, dans R^2 et dans R^3. On procède dans un premier temps à un état de l'art. On se focalise ensuite sur les disques et les boules. En effet, ils font partie des rares formes pour lesquelles il est possible de calculer explicitement et relativement facilement les valeurs propres. On verra malheureusement que ces formes ne sont la plupart du temps pas des minimiseurs. Enfin on s'intéresse aux simulations numériques possibles. En effet, puisque peu de calculs théoriques peuvent être faits il est intéressant d'obtenir numériquement des candidats. Cela permet ensuite d'avoir des hypothèses de travail théorique. `{A} cet effet nous donnerons des éléments de compréhension sur une méthode de simulation numérique ainsi que des résultats obtenus. / The optimization of Laplacian eigenvalues is a classical problem. In fact, at the end of the nineteenth century, Lord Rayleigh conjectured that the first eigenvalue with Dirichlet boundary condition is minimized by a disk. This problem received a lot of attention since this first study and research possibilities are numerous: various conditions, geometrical constraints added, existence, description of optimal shapes... In this document we restrict us to Dirichlet and Neumann boundary conditions in R^2 and R^3. We begin with a state of the art. Then we focus our study on disks and balls. Indeed, these are some of the only shapes for which it is possible to explicitly and relatively easily compute the eigenvalues. But we show in one of the main result of this document that they are not minimizers for most eigenvalues. Finally we take an interest in the possible numerical experiments. Since we can do very few theoretical computations, it is interesting to get numerical candidates. Then we can deduce some theoretical working assumptions. With this in mind we give some keys to understand our numerical method and we also give some results obtained.