Efficient discretization techniques are of crucial importance for most types of problems in numerical mathematics, starting from tasks like how to define sets of points to approximate, interpolate, or integrate certain classes of functions as good as possible, up to 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. In this paper, we discuss two new algorithmic developments concerning the sparse grid finite element discretization of elliptic partial differential equations. First, a method for the numerical treatment of the general linear elliptic differential operator of second order is presented which, with the help of mapping techniques, allows to tackle problems on more complicated geometries. Second, we leave the approximation space of the piecewise multilinear functions and introduce hierarchical polynomial bases of piecewise arbitrary degree that lead to a very straightforward and efficient access to an approximation of higher order on sparse grids. Both algorithms discussed here have been designed in a unidirectional way that allows the recursive reduction of the general d-dimensional case to the simpler 1 D one and, thus, the formulation of programs for arbitrary d.     «

Efficient discretization techniques are of crucial importance for most types of problems in numerical mathematics, starting from tasks like how to define sets of points to approximate, interpolate, or integrate certain classes of functions as good as possible, up to 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...     »