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

Inria, Concace joint team between Airbus CR&T, Cerfacs and Inria, Talence, France

^* Corresponding author: This email address is being protected from spambots. You need JavaScript enabled to view it.

Received: 21 November 2025
Revised: 13 April 2026
Accepted: 24 April 2026

Abstract

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.