Analysis of iterative methods for the viscous/inviscid coupled problem via a spectral element approximation

Based on a new global variational formulation, a spectral element approximation of the incompressible Navier-Stokes/Euler coupled problem gives rise to a global discrete saddle problem. The classical Uzawa algorithm decouples the original saddle problem into two positive definite symmetric systems. Iterative solutions of such systems are feasible and attractive for large problems. Provided an appropriate pre-conditioner is chosen for the pressure system, the nested conjugate gradient methods can be applied to obtain rapid convergence rates. Detailed numerical examples are given to prove the quality of the pre-conditioner. Thanks to the rapid iterative convergence, the global Uzawa algorithm takes advantage of this as compared with the classical iteration by sub-domain procedures. Furthermore, a generalization of the pre-conditioned iterative algorithm to flow simulation is carried out. Comparisons of computational complexity between the Navier-Stokes/Euler coupled solution and the full Navier-Stokes solution are made

[1]  P. Raviart,et al.  Finite Element Approximation of the Navier-Stokes Equations , 1979 .

[2]  K. Horiuti,et al.  Comparison of conservative and rotational forms in large Eddy simulation of turbulent channel flow , 1987 .

[3]  Alessandro Russo,et al.  A self-adaptive formulation for the Euler/Navier—Stokes coupling , 1989 .

[4]  T. A. Zang,et al.  Spectral methods for fluid dynamics , 1987 .

[5]  T. A. Zang,et al.  On the rotation and skew-symmetric forms for incompressible flow simulations , 1991 .

[6]  Anthony T. Patera,et al.  Spectral element methods: Algorithms and architectures , 1988 .

[7]  F. Brezzi.  On the existence, uniqueness and approximation of saddle-point problems arising from lagrangian multipliers , 1974 .

[8]  Einar M. Rønquist,et al.  An Operator-integration-factor splitting method for time-dependent problems: Application to incompressible fluid flow , 1990 .

[9]  Alfio Quarteroni,et al.  A relaxation procedure for domain decomposition methods using finite elements , 1989 .

[10]  Anthony T. Patera,et al.  Analysis of Iterative Methods for the Steady and Unsteady Stokes Problem: Application to Spectral Element Discretizations , 1993, SIAM J. Sci. Comput..

[11]  R. A. Brown,et al.  Multiple buoyancy-driven flows in a vertical cylinder heated from below , 1984, Philosophical Transactions of the Royal Society of London. Series A, Mathematical and Physical Sciences.

[12]  S. Orszag,et al.  Boundary conditions for incompressible flows , 1986 .

[13]  J. Cahouet,et al.  Some fast 3D finite element solvers for the generalized Stokes problem , 1988 .

[14]  Einar M. Rønquist,et al.  Optimal spectral element methods for the unsteady three-dimensional incompressible Navier-Stokes equations , 1988 .

[15]  Andrew J. Wathen,et al.  Fast iterative solution of stabilised Stokes systems, part I: using simple diagonal preconditioners , 1993 .

[16]  Roland Glowinski,et al.  On a mixed finite element approximation of the Stokes problem (I) , 1979 .

[17]  S. Orszag.  Spectral methods for problems in complex geometries , 1980 .

[18]  P. Moin,et al.  Application of a Fractional-Step Method to Incompressible Navier-Stokes Equations , 1984 .

[19]  Paul Fischer,et al.  An Overlapping Schwarz Method for Spectral Element Solution of the Incompressible Navier-Stokes Equations , 1997 .