Spectral analysis and fast methods for large matrices arising from PDE approximation

The main goal of this thesis is to show the crucial role that plays the symbol in analysing the spectrum the sequence of matrices resulting from PDE approximation and in designing a fast method to solve the associated linear problem. In the first part, we study the spectral properties of the matrices arising from $\mathbb{P}_k$ Lagrangian Finite Elements approximation of second order elliptic differential problem with Dirichlet boundary conditions and where the operator is $\mathrm{div} \left(-a(\mathbf{x}) \nabla\cdot\right)$, with $a$ continuous and positive over $\overline \Omega$, $\Omega$ being an open and bounded subset of $\mathbb{R}^d$, $d\ge 1$. We investigate the spectral distribution in the Weyl sense, with a concise overview on localization, clustering, extremal eigenvalues, and asymptotic conditioning. We study in detail the case of constant coefficients on $\Omega=(0,1)^2$ and we give a brief account in the case of variable coefficients and more general domains. While in the second part, we design a fast method of multigrid type for the resolution of linear systems arising from the $\mathbb{Q}_k$ Finite Elements approximation of the same considered problem in one and higher dimensional. The analysis is performed in one dimension, while the numerics are carried out also in higher dimension $d\ge 2$, demonstrating an optimal behavior in terms of the dependency on the matrix size and a robustness with respect to the dimensionality $d$ and to the polynomial degree $k$.