Fast summation by interval clustering for an evolution equation with memory

Details Fast summation by interval clustering for an evolution equation with memory Fast summation by interval clustering for an evolution equation with memory

2012-03-19

arxiv.org/abs/1203.4032v1

Mathematics

Numerical Analysis

28 pages, 1 figure

Scientific paper

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.

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