A posteriori error estimators for hierarchical B-spline discretizations

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.