  Sparse grids are tailored to the approximation of smooth high-dimensional
functions. On a $d$-dimensional tensor product space, the number of grid points
is $N = \mathcal O(h^{-1} |\log h|^{d-1})$, where $h$ is a mesh parameter. The
so-called combination technique, based on hierarchical decomposition and
extrapolation, requires specific multivariate error expansions of the
discretisation error on Cartesian grids to hold. We derive such error
expansions for linear difference schemes through an error correction technique
of semi-discretisations. We obtain overall error formulae of the type $\epsilon
= \mathcal{O} (h^p |\log h|^{d-1})$ and analyse the convergence, with its
dependence on dimension and smoothness, by examples of linear elliptic and
parabolic problems, with numerical illustrations in up to eight dimensions.
