Complexity of hierarchical refinement for a class of admissible mesh configurations

TL;DR

This paper analyzes the complexity of hierarchical refinement in adaptive isogeometric methods, providing estimates on mesh growth relative to marked elements, extending finite element complexity results to spline-based approaches.

Contribution

It offers the first complexity estimate for hierarchical refinement in admissible mesh configurations within adaptive isogeometric methods.

Findings

Provides a complexity estimate for mesh growth

Extends finite element complexity results to spline-based methods

Offers theoretical bounds for adaptive mesh refinement

Abstract

An adaptive isogeometric method based on $d$-variate hierarchical spline constructions can be derived by considering a refine module that preserves a certain class of admissibility between two consecutive steps of the adaptive loop [6]. In this paper we provide a complexity estimate, i.e., an estimate on how the number of mesh elements grows with respect to the number of elements that are marked for refinement by the adaptive strategy. Our estimate is in the line of the similar ones proved in the finite element context, [3,24].