Numerical Solution via Laplace Transforms of a Fractional Order Evolution Equation

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 coefficients, which may be solved in parallel. Three different methods, using 2N+1 quadrature points, are discussed. The first has an error of order O(e ) away from t = 0, whereas the second and third methods are uniformly accurate of order O(e √ N ). Unlike the first and second methods, the third method does not use the Laplace transform of the forcing term. The basic analysis of the time discretization takes place in a Banach space setting and uses a resolvent estimate for the associated elliptic operator. The methods are combined with finite element discretization in the spatial variables to yield fully discrete methods.