This thesis is devoted to theoretical and experimental justifications of numerical methods for fractional differential equations, which have received significant attention over the past decades due to their extraordinary capability of modeling the dynamics of anomalous diffusion processes. In recent years, a number of numerical schemes were developed, analyzed and tested. However, in most of these interesting works, the error estimates were established under the assumption that the solution is sufficiently smooth. Unfortunately, it has been shown that these assumptions are too restrictive for the solutions of fractional differential equations. The goal of this thesis is to develop robust numerical schemes and to establish optimal error bounds that are expressed directly in terms of the regularity of the problem data. We are especially interested in the case of nonsmooth data arising in many applications, e.g. inverse and control problems. After some background introduction and preliminaries in Chapters 1 and 2, we analyze two semidiscrete schemes obtained by standard Galerkin finite element approximation and lumped mass finite element method in Chapter 3. The error bounds for approximate solutions of the homogeneous and inhomogeneous problems are established separately in terms of the smoothness of the data directly. In Chapter 4 we revisit the most popular fully discrete scheme based on L1-type approximation in time and Galerkin finite element method in space and show the first order convergence in time by the discrete Laplace transform technique, which fills the gap between the existing error analysis theory and numerical results. Two fully discrete schemes based on convolution quadrature are developed in Chapter 5. The error bounds are given using two different techniques, i.e., discretized operational calculus and discrete Laplace transform. Last, in Chapter 6, we summarize our work and mention possible future research topics. In each chapter, the discussion is focused on the fractional diffusion model and then extended to some other fractional models. Throughout, numerical results for one- and two-dimensional examples will be provided to illustrate the convergence theory.
