Refined a posteriori error estimation for classical and pressure-robust Stokes finite element methods P. L. LEDERER Institute for Analysis and Scientific Computing, TU Wien C. MERDON Weierstrass Institute for Applied Analysis and Stochastics, Berlin J. SCH ¨ OBERL Institute for Analysis and Scientific Computing, TU Wien Abstract. Recent works showed that pressure-robust modifications of mixed finite element meth- ods for the Stokes equations outperform their standard versions in many cases. This is achieved by divergence-free reconstruction operators and results in pressure independent velocity error estim- ates which are robust with respect to small viscosities. In this paper we develop a posteriori error control which reflects this robustness. The main difficulty lies in the volume contribution of the standard residual-based approach that includes the L 2 -norm of the right-hand side. However, the velocity is only steered by the divergence- free part of this source term. An efficient error estimator must approximate this divergence-free part in a proper manner, otherwise it can be dominated by the pressure error. To overcome this difficulty a novel approach is suggested that uses arguments from the stream function and vorticity formulation of the Navier–Stokes equations. The novel error estimators only take the curl of the right-hand side into account and so lead to provably reliable, efficient and pressure-independent upper bounds in case of a pressure-robust method in particular in pressure- dominant situations. This is also confirmed by some numerical examples with the novel pressure- robust modifications of the Taylor–Hood and mini finite element methods. incompressible Navier–Stokes equations and mixed finite elements and pressure robustness and a posteriori error estimators and adaptive mesh refinement 1. Introduction This paper studies a posteriori error estimators for the velocity of the Stokes equation with a special focus on pressure-robust finite element methods. Pressure-robustness is closely related to the L 2 -orthogonality of divergence-free functions onto gradients of H 1 -functions. In particular, the exact velocity u of the Stokes equations (with zero boundary data), ´ν Δu ` ∇p “ f in Ω and u P V 0 :“tv P H 1 0 pΩq 2 : divv “ 0u, is orthogonal onto any q P L 2 pΩq in the sense that ş Ω qdivpuq dx “ 0. Consequently, u also solves the Stokes equations with f replaced by f ` ∇q for q P H 1 pΩq. This invariance property is in general not preserved for discretely divergence-free testfunctions of most classical finite element methods that E-mail addresses: philip.lederer@tuwien.ac.at, christian.merdon@wias-berlin.de, joachim.schoeberl@tuwien.ac.at. Date : 5th of December, 2017. 1 arXiv:1712.01625v1 [math.NA] 5 Dec 2017