Analysis of Linear Difference Schemes in the Sparse Grid Combination Technique
نویسنده
چکیده
Sparse grids are tailored to the approximation of smooth high-dimensional functions. On a d-dimensional cube, the number of grid points is N = O(h−1| log h|d−1) with a mesh size parameter h. The so-called combination technique, based on hierarchical decomposition, facilitates the numerical solution of partial differential equations on these grids. Key to the convergence analysis are specific multivariate error expansions, which we derive in a generic way for linear difference schemes through an error correction technique employing semi-discretisations. We obtain error formulae of the structure ε = O(hp| log h|d−1) and illustrate the convergence by numerical examples.
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