A posteriori error estimators for hierarchical B-spline discretizations

In this paper, we develop a function-based a posteriori error estimators for the solution of linear second-order elliptic problems considering hierarchical spline spaces for the Galerkin discretization. We obtain a global upper bound for the energy error over arbitrary hierarchical mesh configurations which simplifies the implementation of adaptive refinement strategies. The theory hinges on some weighted Poincaré-type inequalities where the B-spline basis functions are the weights appearing in the norms. Such inequalities are derived following the lines in [A. Veeser and R. Verfürth, Explicit upper bounds for dual norms of residuals, SIAM J. Numer. Anal. 47 (2009) 2387-2405], 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 the discretization, together with an adaptive algorithm guided by these local estimators that yields optimal meshes and rates of convergence, exhibiting an excellent performance.