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

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.