Concepts for higher order finite elements on sparse grids

On the way to an efficient implementation of finite element algorithms related to the pand h-p-versions on sparse grids, we present a general concept for the construction of hierarchical bases of higher order suitable for sparse grid methods. For the solution of partial differential equations, this approach allows us to profit both from the efficiency of sparse grid discretizations and from the advantages of higher order basis functions with regard to their approximation accuracy. We discuss the general relations of sparse grids and higher order techniques, and we report the results of some first numerical experiments concerning piecewise biquadratic hierarchical basis functions.