On the construction of sparse tensor product spaces

Abstract

Let Ω₁ ⊂ &#8477;^{<em>n</em>₁} and Ω₂ ⊂ &#8477;^{<em>n</em>₂} be two given domains and consider on each domain a multiscale sequence of ansatz spaces of polynomial exactness <em>r</em>₁and <em>r</em>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 Ω₁ × Ω₂. Numerical results validate our theoretical findings.