Publikationen SGS: Bibliographie 1997 BibTeX
@article {ART-1997-11,
author = {Hans-Joachim Bungartz},
title = {{A multigrid algorithm for higher order finite elements on sparse grids}},
journal = {Electronic Transactions on Numerical Analysis},
publisher = {Kent State University},
volume = {6},
pages = {63--77},
type = {Artikel in Zeitschrift},
month = {Dezember},
year = {1997},
isbn = {1068-9613},
language = {Englisch},
cr-category = {G.1 Numerical Analysis},
contact = {Hans-Joachim Bungartz bungartz@ipvs.uni-stuttgart.de},
department = {Universit{\"a}t Stuttgart, Institut f{\"u}r Parallele und Verteilte Systeme, Simulation gro{\ss}er Systeme},
abstract = {For most types of problems in numerical mathematics, efficient discretization
techniques are of crucial importance. This holds for tasks like how to define
sets of points to approximate, interpolate, or integrate certain classes of
functions as accurate as possible as well as for the numerical solution of
differential equations. Introduced by Zenger in 1990 and based on hierarchical
tensor product approximation spaces, sparse grids have turned out to be a very
efficient approach in order to improve the ratio of invested storage and
computing time to the achieved accuracy for many problems in the areas
mentioned above. Concerning the sparse grid finite element discretization of
elliptic partial differential equations, recently, the class of problems that
can be tackled has been enlarged significantly. First, the tensor product
approach led to the formulation of unidirectional algorithms which are
essentially independent of the number d of dimensions. Second, techniques for
the treatment of the general linear elliptic differential operator of second
order have been developed, which, with the help of domain transformation,
enable us to deal with more complicated geometries, too. Finally, the
development of hierarchical polynomial bases of piecewise arbitrary degree p
has opened the way to a further improvement of the order of approximation. In
this paper, we discuss the construction and the main properties of a class of
hierarchical polynomial bases and present a symmetric and an asymmetric finite
element method on sparse grids, using the hierarchical polynomial bases for
both the approximation and the test spaces or for the approximation space only,
resp., with standard piecewise multilinear hierarchical test functions. In both
cases, the storage requirement at a grid point does not depend on the local
polynomial degree p, and p and the resulting representations of the basis
functions can be handled in an efficient and adaptive way. An advantage of the
latter approach, however, is the fact that it allows the straightforward
implementation of a multigrid solver for the resulting system which is
discussed, too.},
url = {http://www2.informatik.uni-stuttgart.de/cgi-bin/NCSTRL/NCSTRL_view.pl?id=ART-1997-11&engl=0}
}
