@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 = {Article in Journal},
   month = {December},
   year = {1997},
   isbn = {1068-9613},
   language = {English},
   cr-category = {G.1 Numerical Analysis},
   contact = {Hans-Joachim Bungartz bungartz@ipvs.uni-stuttgart.de},
   department = {University of Stuttgart, Institute of Parallel and Distributed Systems, Simulation of Large Systems},
   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=1}
}