On the construction of sparse tensor product spaces

Let Ω1 ⊂ Rn1 and Ω2 ⊂ Rn2 be two given domains and consider on each domain a multiscale sequence of ansatz spaces of polynomial exactness r1 and r2, 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.