This thesis consists of three chapters, first of which is about a dir=rtl align=right We study a fractional diffusion equation, involving a parameter in the range -1 ? 0, where the fractional power is the Riemann–Liouville sense. The problem provides a macroscopic, continuum model of sub-diffusion, with u giving the density of the diffusing particles that have mean-square displacement proportional to t 1+? . In the limiting case ?=0, we have Brownian motion, and u obeys the dir=rtl align=right In [27], McLean and Mustapha Kassem studied discontinuous Galerkin methods for the time discretization of problem in the case 0 ? 1 and proving optimal error bounds for piecewise-constant and and piecewise-linear trial functions. The convergence analysis for the case -1 ? 1 is more difficult. In the second chapter, we first employ a piecewise-constant discontinuous Galerkin method for the time discretization of a sub-diffusion equation. Thus, our scheme is essentially a modified implicit Euler method. In fact, we prove sub-optimal convergence of order k 1+ ? , where k denotes the maximum time-step, assuming suitable control of Au'. Subsequently, we show an optimal convergence rate of order k, but with Au'' appearing in the error bound. Since Au'(t) and Au''(t) are typically singular at t=0, to achieve the convergence rates cited above, we must employ non-uniform time steps in practice. The formulation of our numerical method allows for a spatial discretization by a Galerkin method using a trial space in D(A 1/2 ), the domain of A 1/2 , where A is a self-adjoint operator. For instance, our approximate solution U (x,t) may be continuous and piecewise-linear in x, but discontinuous and piecewise-constant in t. We find in Theorem 3.2.1 that the method is unconditionally stable even when we choose a different trial space for each time step combined with arbitrarily-spaced time levels. This robustness is a significant advantage of the discontinuous Galerkin method and allowsgreat flexibility in the choice of mesh. We also show that a spatial discretization using continuous piecewise-linear finite elements leads to an additional error term of order h 2 max(1,log k -1 ). Furthermore, we use a piecewise-linear discontinuous Galerkin method for the time discretization of a sub-diffusion equation. Our analysis shows that for a time interval (0,T) and a spatial domain ?, the error in L ? ((0,T) ; L 2 (?)) is of order k 2+ ? . We also consider a fully-discrete scheme that employs standard (continuous) piecewise-linear finite elements in space, and show that the additional error is of order h 2 log(1/k). Numerical experiments indicate that our O(k 2+ ? ) error bound is pessimistic. Then, we show that the error is of order k ? , where ?=min(2 , ). Here, if ? ? ? 0, then we have optimal O(k 2 ) convergence. Moreover, the analysis also shows an improved, but still suboptimal, O( ) error bound when -1? ? ? .