A multiscale sparse grid technique for a
two-dimensional convection-diffusion problem
with exponential layers.
Stephen Russell and Niall
Madden.
Abstract
We investigate the application of a multiscale sparse grid finite element
method for computing numerical solutions to a two-dimensional
singularly
perturbed convection-diffusion problem posed on the unit square. Typically, sparse
grid methods are constructed using a hierarchical basis (see, e.g., Bungartz and
Griebel [1]). In our approach, the method is presented as a
generalisation of the two-scale method described in Liu et al. [3],
and is related to the combination technique outlined by Pflaum and
Zhou [7]. We show that this method retains the same level of
accuracy, in the energy norm, as both the standard Galerkin and
two-scale methods. The computational cost associated with the method,
however, is $\mathcal{O}(N \log N)$, compared to $\mathcal{O}(N^2)$
and $\mathcal{O}(N^{3/2})$ for the Galerkin and two-scale methods respectively.