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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.
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