On the construction of sparse tensor product spaces

Abstract

Let Ω 1 ⊂ R n 1 \Omega _1\subset \mathbb {R}^{n_1} and Ω 2 ⊂ R n 2 \Omega _2\subset \mathbb {R}^{n_2} be two given domains and consider on each domain a multiscale sequence of ansatz spaces of polynomial exactness r 1 r_1 and r 2 r_2 , respectively. In this paper, we study the optimal construction of sparse tensor products made from these spaces. In particular, we derive the resulting cost complexities to approximate functions with anisotropic and isotropic smoothness on the tensor product domain Ω 1 × Ω 2 \Omega _1\times \Omega _2 . Numerical results validate our theoretical findings.