On Newton-Krylov multigrid methods for the imcompressible Navier-Stokes equations
Journal of Computational Physics
A Multigrid-Preconditioned Newton--Krylov Method for the Incompressible Navier--Stokes Equations
SIAM Journal on Scientific Computing
Numerical Methods for Unconstrained Optimization and Nonlinear Equations (Classics in Applied Mathematics, 16)
Numerical Study of a Modified Time-Stepping θ-Scheme for Incompressible Flow Simulations
Journal of Scientific Computing
Fast iterative solvers for buoyancy driven flow problems
Journal of Computational Physics
New robust nonconforming finite elements of higher order
Applied Numerical Mathematics
Newton multigrid least-squares FEM for the V-V-P formulation of the Navier-Stokes equations
Journal of Computational Physics
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We present special numerical simulation methods for non-isothermal incompressible viscous fluids which are based on LBB-stable FEM discretization techniques together with monolithic multigrid solvers. For time discretization, we apply the fully implicit Crank-Nicolson scheme of 2nd order accuracy while we utilize the high order Q"2P"1 finite element pair for discretization in space which can be applied on general meshes together with local grid refinement strategies including hanging nodes. To treat the nonlinearities in each time step as well as for direct steady approaches, the resulting discrete systems are solved via a Newton method based on divided differences to calculate explicitly the Jacobian matrices. In each nonlinear step, the coupled linear subproblems are solved simultaneously for all quantities by means of a monolithic multigrid method with local multilevel pressure Schur complement smoothers of Vanka type. For validation and evaluation of the presented methodology, we perform the MIT benchmark 2001 [M.A. Christon, P.M. Gresho, S.B. Sutton, Computational predictability of natural convection flows in enclosures, in: First MIT Conference on Computational Fluid and Solid Mechanics, vol. 40, Elsevier, 2001, pp. 1465-1468] of natural convection flow in enclosures to compare our results with respect to accuracy and efficiency. Additionally, we simulate problems with temperature and shear dependent viscosity and analyze the effect of an additional dissipation term inside the energy equation. Moreover, we discuss how these FEM-multigrid techniques can be extended to monolithic approaches for viscoelastic flow problems.