  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 $k$.
