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

TL;DR

This paper develops and analyzes sparse-grid B-spline approximation spaces on general geometric domains, demonstrating their efficiency and equivalence to tensor product spaces with fewer degrees of freedom, under certain regularity conditions.

Contribution

It introduces two constructions of sparse-grid B-spline spaces on mapped domains and proves their mathematical equivalence, providing error estimates and highlighting their advantages over standard tensor products.

Findings

Sparse-grid B-spline spaces achieve the same approximation order as tensor products with fewer degrees of freedom.

Error estimates show advantages of sparse-grid tensor products in approximation accuracy.

Stronger regularity assumptions are needed for optimal convergence on non-tensor-product domains.

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.