Authors: Afaq, Muhammad Aaqib
Fatima, Arooj
Turek, Stefan
Ouazzi, Abderrahim
Title: Robust Monolithic Multigrid FEM Solver for Three Field Formulation of Incompressible Flow Problems
Language (ISO): en
Abstract: Numerical simulation of three field formulations of incompressible flow problems is of interest for many industrial applications, for instance macroscopic modeling of Bing-ham, viscoelastic and multiphase flows, which usually consists in supplementing the mass and momentum equations with a differential constitutive equation for the stress field. The variational formulation rising from such continuum mechanics problems leads to a three field formulation with saddle point structure. The solvability of the problem requires different compatibility conditions (LBB conditions) [1] to be satisfied. Moreover, these constraints over the choice of the spaces may conflict/challenge the robustness and the efficiency of the solver. For illustrating the main points, we will consider the three field formulation of the Navier-Stokes problem in terms of velocity, stress, and pressure. Clearly, the weak form imposes the compatibility constraints over the choice of velocity, stress, and pressure spaces. So far, the velocity-pressure combi-nation took much more attention from the numerical analysis and computational fluid dynamic community, which leads to some best interpolation choices for both accuracy and efficiency, as for instance the combination Q2/P1disc. To maintain the computational advantages of the Navier-Stokes solver in two field formulations, it may be more suitable to have a Q2 interpolation for the stress as well, which is not stable in the absence of pure viscous term [2]. We proceed by adding an edge oriented stabilization to overcome such situation. Furthermore, we show the robustness and the efficiency of the resulting discretization in comparison with the Navier-Stokes solver both in two field as well as in three field formulation in the presence of pure viscous term. Moreover, the benefit of adding the edge oriented finite element stabilization (EOFEM) [3, 4] in the absence of the pure viscous term is tested. The nonlinearity is treated with a Newton-type solver [5] with divided difference evaluation of the Jacobian matrices [6, 7]. The resulting linearized system inside of the outer Newton solver is a typical saddle point problem which is solved using a geometrical multigrid method with Vanka-like smoother [8, 9]. The method is implemented into the FeatFlow [10] software package for the numerical simulation. The stability and robustness of the method is numerically investigated for ”flow around cylinder” benchmark [7, 11].
Subject Headings: finite element method
edge oriented stabilization
Babuska-Brezzi conditions
Navier-Stokes equations
Issue Date: 2023-01
Appears in Collections:Ergebnisberichte des Instituts für Angewandte Mathematik

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