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We consider isogeometric discretizations of the Poisson model problem,focusing on high polynomial degrees and strong hierarchical refinements. Wederive a posteriori error estimates by equilibrated fluxes, i.e., vector-valuedmapped piecewise polynomials lying in the $\boldsymbol{H}({\rm div})$ spacewhich appropriately approximate the desired divergence constraint. Ourestimates are constant-free in the leading term, locally efficient, and robustwith respect to the polynomial degree. They are also robust with respect to thenumber of hanging nodes arising in adaptive mesh refinement employinghierarchical B-splines. Two partitions of unity are designed, one with largersupports corresponding to the mapped splines, and one with small supportscorresponding to mapped piecewise multilinear finite element hat basisfunctions. The equilibration is only performed on the small supports, avoidingthe higher computational price of equilibration on the large supports or eventhe solution of a global system. Thus, the derived estimates are also asinexpensive as possible. An abstract framework for such a setting is developed,whose application to a specific situation only requests a verification of a fewclearly identified assumptions. Numerical experiments illustrate thetheoretical developments.

Inexpensive polynomial-degree-robust equilibrated flux a posteriori estimates for isogeometric analysis