@Article{ Griebel.Oeltz.Vassilevski:2005,
author = {Michael Griebel and Daniel Oeltz and Panayot Vassilevski},
title = {Space-time approximation with Sparse Grids},
institution = {Lawrence Livermore National Laboratory},
year = {2005},
journal = {SIAM J. Sci. Comput.},
volume = {28},
number = {2},
pages = {701-727},
pdf = {http://wissrech.ins.uni-bonn.de/research/pub/oeltz/IM-319022-3-preprint.pdf}
,
annote = {article,C2},
abstract = {In this report we introduce approximation spaces for
parabolic problems which are based on the tensor product
construction of a multiscale basis in space and a
multiscale basis in time. Proper truncation then leads to
so-called space-time sparse grid spaces. For a uniform
discretization of the spatial space of dimension d with
O(N^d) degrees of freedom, these spaces involve for d > 1
also only O(N^d) degrees of freedom for the discretization
of the whole space-time problem. But they provide the same
approximation rate as classical space-time Finite Element
spaces which need O(N^(d+1)) degrees of freedoms. This
makes these approximation spaces well suited for
conventional parabolic and for time-dependent optimization
problems.
We analyze the approximation properties and the dimension
of these sparse grid space-time spaces for general stable
multiscale bases. We then restrict ourselves to an
interpolatory multiscale basis, i.e. a hierarchical basis.
Here, to be able to handle also complicated spatial domains
Omega, we construct the hierarchical basis from a given
spatial Finite Element basis as follows: First we determine
coarse grid points recursively over the levels by the
coarsening step of the algebraic multigrid method. Then, we
derive interpolatory prolongation operators between the
respective coarse and fine grid points by a least square
approach. This way we obtain an algebraic hierarchical
basis for the spatial domain which we then use in our
space-time sparse grid approach.
We give numerical results on the convergence rate of the
interpolation error of these spaces for various space-time
problems with two spatial dimensions. Also implementational
issues, data structures and questions of adaptivity are
addressed to some extent.},
note = {Also as Preprint No.~222, SFB 611, University of Bonn}
}