On the convergence of the combination technique

Sparse tensor product spaces provide an efficient tool to discretize higher dimensional operator equations. The direct Galerkin method in such ansatz spaces may employ hierarchical bases, interpolets, wavelets or multilevel frames. Besides, an alternative approach is provided by the so-called combination technique. It properly combines the Galerkin solutions of the underlying problem on certain full (but small) tensor product spaces. So far, however, the combination technique has been analyzed only for special model problems. In the present paper, we provide now the analysis of the combination technique for quite general operator equations in sparse tensor product spaces. We prove that the combination technique produces the same order of convergence as the Galerkin approximation with respect to the sparse tensor product space. Furthermore, the order of the cost complexity is the same as for the Galerkin approach in the sparse tensor product space. Our theoretical findings are validated by numerical experiments.