A posteriori error estimators for hierarchical B-spline discretizations

TL;DR

This paper develops and analyzes function-based a posteriori error estimators for hierarchical B-spline discretizations of elliptic problems, providing theoretical bounds and demonstrating efficiency through numerical experiments.

Contribution

It introduces new a posteriori error estimators for hierarchical B-spline spaces with proven bounds and adaptive algorithms that achieve optimal convergence rates.

Findings

Error estimators are effective regardless of spline degree.

Adaptive algorithms produce optimal meshes and convergence rates.

Numerical results confirm the theoretical efficiency of the estimators.

Abstract

In this article we develop function-based a posteriori error estimators for the solution of linear second order elliptic problems considering hierarchical spline spaces for the Galerkin discretization. We prove a global upper bound for the energy error. The theory hinges on some weighted Poincar\'e type inequalities, where the B-spline basis functions are the weights appearing in the norms. Such inequalities are derived following the lines in [Veeser and Verf\"urth, 2009], where the case of standard finite elements is considered. Additionally, we present numerical experiments that show the efficiency of the error estimators independently of the degree of the splines used for discretization, together with an adaptive algorithm guided by these local estimators that yields optimal meshes and rates of convergence, exhibiting an excellent performance.