Local power of approximation in hierarchical spline spaces on weakly admissible meshes

TL;DR

This paper investigates local approximation in hierarchical spline spaces on weakly admissible meshes, introducing a robust adaptive refinement algorithm and establishing stability and approximation results.

Contribution

It provides the first rigorous analysis of weakly admissible hierarchical meshes with nested cell sets for local spline approximation.

Findings

The proposed adaptive refinement algorithm effectively constructs locally graded meshes.

Stability and approximation properties are rigorously established for the hierarchical spline spaces.

Numerical results show the approach outperforms existing adaptive strategies.

Abstract

We study local approximation properties in hierarchical spline spaces through a twofold approach. First, we design and analyze a robust adaptive refinement algorithm to construct locally graded meshes. Second, we establish rigorous stability and approximation results using computationally efficient quasi-interpolation operators. The primary contribution is the analysis of weakly admissible hierarchical meshes. Our framework relies on strictly nested cell sets that locally reproduce the full tensor-product spline space at each level. Theoretical and numerical results demonstrate that this intuitive approach is mathematically elegant and outperforms existing adaptive refinement strategies in various practical scenarios.