Title: A comparative study of mixed least-squares FEMs for the incompressible Navier-Stokes equations

Authors: Alexander Schwarz; Masoud Nickaeen; Serdar Serdas; Carina Nisters; Abderrahim Ouazzi; Jörg Schröder; Stefan Turek

Addresses: Faculty of Engineering, Institute for Mechanics, University Duisburg-Essen, Universitätsstr. 15, 45141 Essen, Germany ' Department of Mathematics, Institute for Applied Mathematics, Dortmund University of Technology, Vogelpothsweg 87, 44227 Dortmund, Germany; Richard D. Berlin Center for Cell Analysis and Modeling, University of Connecticut Health Center, Farmington, CT 06030, USA ' Faculty of Engineering, Institute for Mechanics, University Duisburg-Essen, Universitätsstr. 15, 45141 Essen, Germany ' Faculty of Engineering, Institute for Mechanics, University Duisburg-Essen, Universitätsstr. 15, 45141 Essen, Germany ' Department of Mathematics, Institute for Applied Mathematics, Dortmund University of Technology, Vogelpothsweg 87, 44227 Dortmund, Germany ' Faculty of Engineering, Institute for Mechanics, University Duisburg-Essen, Universitätsstr. 15, 45141 Essen, Germany ' Department of Mathematics, Institute for Applied Mathematics, Dortmund University of Technology, Vogelpothsweg 87, 44227 Dortmund, Germany

Abstract: In the present contribution, we compare (quantitatively) different mixed least-squares finite element methods (LSFEMs) with respect to computational costs and accuracy. Various first-order systems are derived based on the residual forms of the equilibrium equation and the continuity condition. The first formulation under consideration is a div-grad first-order system resulting in a three-field formulation with total stresses, velocities, and pressure (S-V-P) as unknowns. Here, the variables are approximated in H(div) × H¹ × L² on triangles and in H¹ × H¹ × L² on quadrilaterals. In addition to that a reduced stress-velocity (S-V) formulation is derived and investigated. S-V-P and S-V formulations are promising approaches when the stresses are of special interest, e.g., for non-Newtonian, multiphase or turbulent flows. The main focus of the work is drawn to performance and accuracy aspects on the one side for finite elements with different interpolation orders and on the other side on the usage of efficient solvers, for instance of Krylov-space or multigrid type.

Keywords: least-squares FEM; V-V-P formulation; S-V-P formulation; S-V formulation; Navier-Stokes; multigrid.

DOI: 10.1504/IJCSE.2018.094421

International Journal of Computational Science and Engineering, 2018 Vol.17 No.1, pp.80 - 97

Received: 11 May 2015
Accepted: 05 Apr 2016
Published online: 03 Sep 2018 *

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