[1] 
M. Griebel and D. Oeltz.
A sparse grid spacetime discretization scheme for parabolic
problems.
Computing, 81(1):134, 2007. [ bib  .ps.gz 1 ] In this paper we consider the discretization in space and time of parabolic differential equations where we use the socalled spacetime sparse grid technique. It employs the tensor product of a onedimensional multilevel basis in time and a proper multilevel basis in space. This way, the additional order of complexity of a direct spacetime discretization can be avoided, provided that the solution fulfills a certain smoothness assumption in spacetime, namely that its mixed spacetime derivatives are bounded. This holds in many applications due to the smoothing properties of the propagator of the parabolic PDE (heat kernel). In the more general case, the spacetime sparse grid approach can be employed together with adaptive refinement in space and time and then leads to similar approximation rates as the nonadaptive method for smooth functions. We analyze the properties of different spacetime sparse grid discretizations for parabolic differential equations from both, the theoretical and practical point of view, discuss their implementational aspects and report on the results of numerical experiments.
