Error Estimates for Sparse Tensor Products of B-spline Approximation Spaces

This work introduces and analyzes B-spline approximation spaces defined on general geometric domains obtained through a mapping from a parameter domain. These spaces are constructed as sparse-grid tensor products of univariate spaces in the parameter domain and are mapped to the physical domain via a geometric parametrization. Both the univariate approximation spaces and the geometric mapping are built using maximally smooth B-splines. We construct two such spaces, employing either the sparse-grid combination technique or the hierarchical subspace decomposition of sparse-grid tensor products, and we prove their mathematical equivalence. Furthermore, we derive approximation error estimates and inverse inequalities that highlight the advantages of sparse-grid tensor products. Specifically, under suitable regularity assumptions on the solution, these spaces achieve the same approximation order as standard tensor product spaces while using significantly fewer degrees of freedom. Additionally, our estimates indicate that, in the case of non-tensor-product domains, stronger regularity assumptions on the solution -- particularly concerning isotropic (non-mixed) derivatives -- are required to achieve optimal convergence rates compared to sparse-grid methods defined on tensor-product domains.