Goal-adaptive Isogeometric Analysis with hierarchical splines

•A goal-oriented error estimator using a p+1 spline dual discretization is proposed.•Two refinement indicators suitable for hierarchical splines are developed.•Two problems with singular primal and dual solutions are adaptively discretized.•Optimal convergence rates are observed in terms of the chosen quantities of interest.

In this work, a method of goal-adaptive Isogeometric Analysis is proposed. We combine goal-oriented error estimation and adaptivity with hierarchical B-splines for local h-refinement. The goal-oriented error estimator is computed with a p-refined discrete dual space, which is adaptively refined alongside the primal space. This discrete dual space is proven to be a strict superset of the primal space. Hierarchical refinements are introduced in marked regions that are formed as the union of chosen coarse-level spline supports from the primal basis. We present two ways of extracting localized refinement indicators suitable for the hierarchical refinement procedure: one based on a partitioning of the dual-weighted residual into contributions of basis function supports and one based on the combination of element indicators within a basis function support. The proposed goal-oriented adaptive strategy is exemplified for the Poisson problem and a free-surface flow problem. Numerical experiments on these problems show convergence of the adaptive method with optimal rates. Furthermore, the corresponding goal-oriented error estimators are shown to be accurate, with effectivity indices in the range of 0.7–1.1.