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

Clément Guillet

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.

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

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.
Paper Structure (21 sections, 17 theorems, 128 equations)

This paper contains 21 sections, 17 theorems, 128 equations.

Table of Contents

  1. Introduction
  2. The Main Results
  3. Preliminaries
  4. Sobolev spaces and norms
  5. Regular tensor-product approximation spaces
  6. Parameter space meshes
  7. Univariate B-splines spaces
  8. Anisotropic $d$-dimensional B-spline spaces
  9. Sparse-grid tensor-product approximation spaces
  10. Hierarchical subspace decomposition
  11. Sparse-grid combination technique
  12. Equivalence between spaces
  13. Error estimate and inverse inequality
  14. Approximation error estimate
  15. Inverse inequality
  16. ...and 6 more sections

Key Result

Theorem 1.1

Let $u \in H^{dq}(\Omega)$ with $q \in \mathbb{N}_0$, then for all $r \in \mathbb{N}_0$ satisfying $0 \leq r \leq q \leq p+1$, there exists a spline approximation $u_{p,h}^{(1)}\in S_{p,h}^{(1)}(\Omega)$ such that

Theorems & Definitions (43)

  • Theorem 1.1
  • Theorem 1.2
  • Definition 3.1
  • Lemma 3.1
  • Definition 3.2
  • Lemma 3.2
  • Definition 3.3
  • Definition 3.4
  • Definition 3.5
  • Proposition 3.1
  • ...and 33 more