Structured matrices coming from PDE approximation theory: spectral analysis, spectral symbol and design of fast iterative solvers.

Partial Differential Equations (PDE) are extensively used in Applied Sciences to model real-world problems. The solution u of a PDE is normally not available in closed form, and so it is necessary to approximate it by means of some numerical method. Despite the differences among the various methods, the principle on which all of them are based is essentially the same: they first discretize the PDE by introducing a mesh, related to some discretization parameter n, and then they compute the corresponding numerical solution u_n, which will converge to u when n tends to infinity, i.e., when the mesh is progressively refined. Now, if both the PDE and the numerical method are linear, the computation of u_n reduces to solving a certain linear system A_n * u_n = f_n whose size d_n tends to infinity with n. In addition, the sequence of discretization matrices A_n often enjoys an asymptotic spectral distribution described by a certain matrix-valued function f, which takes values in the space of Hermitian matrices of a certain size s. This means that, for large n, the eigenvalues of A_n are approximately given by a uniform sampling of the eigenvalue functions lambda_i(f), i=1,...,s, over the domain of f. In this situation, f is called the (spectral) symbol of the sequence of matrices A_n. The identification and the study of the symbol are two interesting issues in themselves, because they provide an accurate information about the asymptotic global behavior of the eigenvalues of A_n. In particular, the number s coincides with the number of "branches" that compose the asymptotic spectrum of A_n. However, the knowledge of the symbol f and of its properties is not only interesting in itself, but can also be used for practical purposes. In particular, it can be used to design effective preconditioned Krylov and multigrid solvers for the linear systems associated with A_n. The reason is clear: the convergence properties of preconditioned Krylov and multigrid methods strongly depend on the spectral features of the matrix to which they are applied. Hence, the spectral information provided by the symbol can be conveniently used for designing fast solvers of this kind. The purpose of this thesis is to present some specific examples, of interest in practical applications, in which the above philosophical discussion comes to life. As our model PDE, we consider classical second-order elliptic differential equations. Concerning the numerical methods that we employ for their solution, we make three choices: the classical Qp Lagrangian Finite Element Method (FEM), the Galerkin B-spline Isogeometric Analysis (IgA) and the B-spline IgA Collocation Method. We first identify and study the symbol f that characterizes the asymptotic spectrum of the discretization matrices A_n arising from these approximation techniques. Then, by exploiting the properties of the symbol, we design fast iterative solvers for the matrices A_n associated with the two numerical methods based on IgA (the Galerkin B-spline IgA and the B-spline IgA Collocation Method).