We consider an initial-boundary value problem for ∂tu − ∂ −α t ∇ 2 u = f (t), that is, for a fractional diffusion (−1 < α < 0) or wave (0 < α < 1) equation. A numerical solution is found by applying a piecewise-linear, discontinuous Galerkin (DG) method in time combined with a piecewise-linear, conforming finite element method in space. The time mesh is… (More)
We employ a piecewise-constant, discontinuous Galerkin method for the time discretization of a sub-diffusion equation. Denoting the maximum time step by k, we prove an a priori error bound of order k under realistic assumptions on the regularity of the solution. We also show that a spatial dis-cretization using continuous, piecewise-linear finite elements… (More)
The paper proves estimates for the partial derivatives of the solution to a time-fractional diffusion equation, posed over a bounded spatial domain. Such estimates are needed for the analysis of effective numerical methods, particularly since the solution is less regular than in the well-known case of classical diffusion.
In this paper we obtain upper and lower bounds on the spectrum of the stiffness matrix arising from a finite element Galerkin approximation (using nodal basis functions) of a bounded, symmetric bilinear form which is elliptic on a Sobolev space of real index m ∈ [−1, 1]. The key point is that the finite element mesh is required to be neither quasiuniform… (More)
We consider the discretization in time of a fractional order diffusion equation. The approximation is based on a further development of the approach of using Laplace transformation to represent the solution as a contour integral, evaluated to high accuracy by quadrature. This technique reduces the problem to a finite set of elliptic equations with complex… (More)
In a previous paper, McLean and Thomée (to appear), we studied three numerical methods for the discretization in time of a fractional order evolution equation, in a Banach space framework. Each of the methods applied a quadrature rule to a contour integral representation of the solution in the complex plane, where for each quadrature point an elliptic… (More)
We consider an initial value problem for a class of evolution equations incorporating a memory term with a weakly singular kernel bounded by í µí° ¶(í µí±¡ − í µí±) í µí»¼−1 , where 0 < í µí»¼ < 1. For the time discretization we apply the discontinuous Galerkin method using piecewise polynomials of degree at most í µí± − 1, for í µí± = 1 or 2. For the… (More)