Adjoint error estimators and adaptive mesh refinement in Nek5000

The development of adaptive mesh refinement capabilities in the field of computational fluid dynamics is an essential tool for enabling the simulation of larger and more complex physical problems. In this report, we describe recent developments that have been made to enable adaptive mesh refinement in Nek5000 and we validate the method on simple, two–dimensional, steady test cases. We start by describing the modifications brought to Nek5000 that enable the presence of hanging nodes in the mesh. Thanks to this new feature, we can use the h-refinement technique for mesh adaptation, where selected elements are split via quadtree (2D) or octree (3D) structures. Then, two methods are considered to estimate and control the error. The first method is local and based on the spectral properties of the solution on each element. The second method is goal-oriented and takes into account both the local properties of the solution and the global dependence of the error in the solution via the resolution of an adjoint problem. Finally, the use of automatic mesh refinement is demonstrated in Nek5000 on two test cases: the lid-driven cavity at Re = 7, 500 and the flow past a cylinder at Re = 40. Both error estimation methods are compared and are shown to efficiently reduce the number of degrees of freedom required to reach a given tolerance on the solution compared to conforming refinement. Moreover, the gains in terms of mesh generation, accuracy and computational cost are discussed by analysing the convergence of some functional of interest and the evolution of the mesh as refinement proceeds.