GLT-based numerical solvers for Fractional Differential Equations

Fractional differential equations are powerful, versatile models for a wide range of real-world phenomena and are commonly used in many applicative fields. On the downside, analytical solutions are often unavailable or difficult to acquire and handle, while their discretization leads to large, dense, and ill-conditioned linear systems, hard to treat computationally. This Thesis explores the development of numerical methods within the framework of Generalized Locally Toeplitz (GLT) matrix sequences. By using GLT theory to analyze the eigenvalues and singular values of the coefficient matrix sequences, we construct efficient and accurate solvers for various multi-dimensional fractional differential equations, including Krylov and multigrid methods for diffusion equations defined on general convex domains, as well as preconditioning techniques for equations involving the fractional Laplacian. In each scenario, we highlight both the peculiarities characterizing the specific problem and general aspects that may be applied in other contexts, testing the theoretical results through several numerical experiments.