We propose and analyze a time-stepping discontinuous Petrov-Galerkin method combined with the continuous conforming finite element method in space for the numerical solution of time-fractional subdiffusion problems. We prove the existence, uniqueness, and stability of approximate solutions and derive error estimates. To achieve high order convergence rates from the time discretizations, the time mesh is graded appropriately near t = 0 to compensate for the singular (temporal) behavior of the exact solution near t = 0 caused by the weakly singular kernel, but the spatial mesh is quasi uniform. In the L∞((0, T ); L 2 (Ω))-norm, ((0, T ) is the time domain and Ω is the spatial domain); for sufficiently graded time meshes, a global convergence of order k m+α/2 + h r+1 is shown, where 0 < α < 1 is the fractional exponent, k is the maximum time step, h is the maximum diameter of the elements of the spatial mesh, and m and r are the degrees of approximate solutions in time and spatial variables, respectively. Numerical experiments indicate that our theoretical error bound is pessimistic. We observe that the error is of order k m+1 + h r+1 , that is, optimal in both variables.

The asymptotic iteration method is applied, to calculate the angular spheroidal

We propose a piecewise-linear, time-stepping discontinuous Galerkin method to solve numerically a time fractional diffusion equation involving Caputo derivative of order µ ∈ (0, 1) with variable coefficients. For the spatial discretization, we apply the standard piecewise linear continuous Galerkin method. Well-posedness of the fully discrete scheme and error analysis will be shown. For a time interval (0, T ) and a spatial domain Ω, our analysis suggest that the error in L 2 (0, T ), L 2 (Ω) -norm is of order O(k 2− µ 2 + h 2 ) (that is, short by order µ 2 from being optimal in time) where k denotes the maximum time step, and h is the maximum diameter of the elements of the (quasi-uniform) spatial mesh. However, our numerical experiments indicate optimal O(k 2 + h 2 ) error bound in the stronger L ∞ (0, T ), L 2 (Ω) -norm. Variable time steps are used to compensate the singularity of the continuous solution near t = 0.In this paper, we investigate a numerical solution that allows a time discontinuity for solving time fractional diffusion equations with variable diffusivity. Let Ω be a bounded convex polygonal domain in R d (d = 1, 2, 3), with a boundary ∂Ω, and

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