Fast summation by interval clustering for an evolution equation with memory

Abstract

We solve a fractional diffusion equation using a piecewise-constant, discontinuous Galerkin method in time combined with a continuous, piecewise-linear finite element method in space. If there are $N$ time levels and $M$ spatial degrees of freedom, then a direct implementation of this method requires $O(N^2M)$ operations and $O(NM)$ active memory locations, owing to the presence of a memory term: at each time step, the discrete evolution equation involves a sum over \emph{all} previous time levels. We show how the computational cost can be reduced to $O(MN\log N)$ operations and $O(M\log N)$ active memory locations.