  This work focuses on the development of a posteriori error estimates for
fourth-order, elliptic, partial differential equations. In particular, we
propose a novel algorithm to steer an adaptive simulation in the context of
Kirchhoff plates and Kirchhoff-Love shells by exploiting the local refinement
capabilities of hierarchical B-splines. The method is based on the solution of
an auxiliary residual-like variational problem, formulated by means of a space
of localized spline functions. This space is characterized by $C^1$ continuous
B-splines with compact support on each active element of the hierarchical mesh.
We demonstrate the applicability of the proposed estimator to Kirchhoff plates
and Kirchhoff-Love shells by studying several benchmark problems which exhibit
both smooth and singular solutions. In all cases, we obtain optimal asymptotic
rates of convergence for the error measured in the energy norm and an excellent
approximation of the true error.
