# Adaptive IGAFEM with optimal convergence rates: Hierarchical B-splines

## TL;DR

This paper introduces an adaptive isogeometric finite element method using hierarchical B-splines for elliptic PDEs, achieving optimal convergence rates through a novel refinement strategy guided by an a posteriori error estimator.

## Contribution

It develops a new adaptive algorithm for IGAFEM with hierarchical B-splines, proving optimal convergence rates and supporting findings with numerical experiments.

## Key findings

- Linear convergence of the error estimator
- Optimal algebraic convergence rates achieved
- Numerical experiments confirm theoretical results

## Abstract

We consider an adaptive algorithm for finite element methods for the isogeometric analysis (IGAFEM) of elliptic (possibly non-symmetric) second-order partial differential equations in arbitrary space dimension $d\ge2$. We employ hierarchical B-splines of arbitrary degree and different order of smoothness. We propose a refinement strategy to generate a sequence of locally refined meshes and corresponding discrete solutions. Adaptivity is driven by some weighted residual a posteriori error estimator. We prove linear convergence of the error estimator (resp. the sum of energy error plus data oscillations) with optimal algebraic rates. Numerical experiments underpin the theoretical findings.