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Summary: Adaptive mesh refinement is performed in the framework of the spectral element method augmented by approaches to error estimation and control. The $$h$$-refinement technique is used for adapting the mesh, where selected grid elements are split by a quadtree (2D) or octree (3D) structure. Continuity between parent-child elements is enforced by high-order interpolation of the solution across the common faces. Parallel mesh partitioning and grid management respectively, are taken care of by the external libraries ParMETIS and p4est. Two methods are considered for estimating and controlling the error of the solution. The first error estimate is local and based on the spectral properties of the solution on each element. This method gives a local measure of the $$L^2$$-norm of the solution over the entire computational domain. The second error estimate uses the dual-weighted residuals method – it is based on and takes into account both the local properties of the solution and the global dependence of the error in the solution via an adjoint problem. The objective of this second approach is to optimize the computation of a given functional of physical interest. The simulations are performed by using the code Nek5000 and three steady-state test cases are studied: a two-dimensional lid-driven cavity at $$\mathrm{Re} = 7500$$, a two-dimensional flow past a cylinder at $$\mathrm{Re} = 40$$, and a three-dimensional lid-driven cavity at $$\mathrm{Re} = 2000$$ with a moving lid tilted by an angle of 30°. The efficiency of both error estimators is compared in terms of refinement patterns and accuracy on the functional of interest. In the case of the adjoint error estimators, the trend on the error of the functional is shown to be correctly represented up to a multiplicative constant.