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In scientific computing, efficient discretization technqiues are of
crucial importance. While a sophisticated a priori choice of grid
patterns has a quite long tradition for the approximation,
interpolation, and integration of functions, the hierarchical sparse
grid concept was the first approach to combine such structural
considerations with adaptive finite element discretizations for
partial differential equations. The most important property of
sparse grids is certainly the fact that the number of degrees of
freedom necessary to achieve a certain given accuracy does not
depend or depends only on very slightly on the problem's
dimensionality d, which is advantageous especially for problems with
large d.
In this text, we deal with both the theoretical and the algorithmic
extension of the piecewise linear approach used so far to polynomial
bases odf an arbitrary and varying degree. The construction of
suitable hierarchical bases with just one degree of freedom per
element and the generalization of the unidirectional sparse grid
algorithms allow to combine the optimal complexity of the sparse
grid approach with the advantages of both adaptive mesh refinement
and higher order approximation.
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