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.