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.

💡 Research Summary

This paper develops and rigorously analyzes sparse‑grid tensor‑product approximation spaces built from maximally smooth B‑splines, and studies their performance when transferred from a reference (parameter) domain to a general physical domain via a smooth geometric mapping. The authors consider two construction paradigms: the classic sparse‑grid combination technique and a hierarchical subspace decomposition. They prove that both approaches generate the same finite‑dimensional space, denoted (S^{(1)}_{p,h}(\Omega)), thus establishing a unified framework for sparse‑grid isogeometric analysis (IGA).

The groundwork begins with a detailed description of univariate B‑spline spaces (S_{p,h_\ell}(0,1)) of degree (p) and maximal continuity (C^{p-1}). Standard approximation estimates (Lemma 3.1) and inverse inequalities (Lemma 3.2) for these spaces are recalled; both hold with constants independent of the spline degree and mesh size. By taking tensor products of the univariate spaces, anisotropic d‑dimensional spline spaces (S_{p,\mathbf{h}\ell}(\hat\Omega)) are formed on the unit hypercube (\hat\Omega=(0,1)^d). A projection operator (\Pi{p,\mathbf{h}_\ell}) is defined as the composition of univariate (H^r)‑orthogonal projections along each coordinate direction.

Proposition 3.1 provides a mixed‑Sobolev error bound for the full tensor‑product space, but this bound is insufficient for sparse‑grid analysis because it does not capture the cancellation of coarse‑grid errors across dimensions. Lemma 3.3 resolves this by expressing the global projection error as a telescopic sum over subsets of coordinate directions: \