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The main goal of this thesis is to show the crucial role that plays thesymbol in analysing the spectrum the sequence of matrices resulting from PDEapproximation and in designing a fast method to solve the associated linearproblem. In the first part, we study the spectral properties of the matricesarising from $\mathbb{P}_k$ Lagrangian Finite Elements approximation of secondorder elliptic differential problem with Dirichlet boundary conditions andwhere the operator is $\mathrm{div} \left(-a(\mathbf{x}) \nabla\cdot\right)$,with $a$ continuous and positive over $\overline \Omega$, $\Omega$ being anopen and bounded subset of $\mathbb{R}^d$, $d\ge 1$. We investigate thespectral distribution in the Weyl sense, with a concise overview onlocalization, clustering, extremal eigenvalues, and asymptotic conditioning. Westudy in detail the case of constant coefficients on $\Omega=(0,1)^2$ and wegive a brief account in the case of variable coefficients and more generaldomains. While in the second part, we design a fast method of multigrid typefor the resolution of linear systems arising from the $\mathbb{Q}_k$ FiniteElements approximation of the same considered problem in one and higherdimensional. The analysis is performed in one dimension, while the numerics arecarried out also in higher dimension $d\ge 2$, demonstrating an optimalbehavior in terms of the dependency on the matrix size and a robustness withrespect to the dimensionality $d$ and to the polynomial degree $k$.

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