A proof of the consistency of the finite difference technique on sparse grids
The author shows the equivalence of the finite discretization of differential operators on sparse grids and the extrapolation formula of the combination technique applied to discretized differential operators on rectangular subgrids which correspond to the subspace \(V^I\) for \(I\subset\Lambda\), where \(\Lambda\) is some index set. This subspace consists of the (anisotropic) tensor product of the space \(V^l= \text{span}\{\varphi(2^l x-s)\mid s\in \mathbb Z\}\), where the scaling function \(\phi\) induces a multiresolution analysis. The combination technique extrapolates a very accurate solution on the orthogonal sum of orthogonal complements of \(V^I\), from finite difference or Galerkin solutions on \(V^I\). The above result gives a new proof of the consistency of the sparse grid finite difference technique, thus improving previous results in the following points: weaker smoothness is required, the proof allows for quite general index set \(\Lambda\) and covers higher order discretization and compact finite difference schemes.
scientific article; zbMATH DE number 1563392