Extrapolation, combination, and sparse grid techniques for elliptic boundary value problems

Several variants of extrapolation can be used for the solution of partial differential equations. There are Richardson extrapolation, truncation error extrapolation, and extrapolation of the functional. In multi-dimensional problems, multivariate error expansions can be exploited by multivariate extrapolation, where the asymptotic expansions in different mesh parameters are exploited. Particularly interesting cases are the combination technique that uses all the grids that have a constant product of the meshspacings in the different coordinate directions. Another related technique is the sparse grid finite element technique that can be interpreted as a combination extrapolation of the functional.