Analysis of Iterative Methods for the Steady and Unsteady Stokes Problem: Application to Spectral Element Discretizations

A new and detailed analysis of the basic Uzawa algorithm for decoupling of the pressure and the velocity in the steady and unsteady Stokes operator is presented. The paper focuses on the following new aspects: explicit construction of the Uzawa pressure-operator spectrum for a semiperiodic model problem; general relationship of the convergence rate of the Uzawa procedure to classical inf-sup discretization analysis; and application of the method to high-order variational discretization.

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

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

[3]  G. Karniadakis,et al.  Recent advances in parallel spectral element simulation of unsteady incompressible flows , 1988 .

[4]  H. Vandeven,et al.  A multidomain spectral collocation method for the Stokes problem , 1990 .

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

[6]  Anthony T. Patera,et al.  A well-posed optimal spectral element approximation for the Stokes problem , 1987 .

[7]  Michael Vogelius,et al.  Divergence stability in connection with the $p$ -version of the finite element method , 1990 .

[8]  Claudio Canuto,et al.  Generalized Inf-Sup Conditions for Chebyshev Spectral Approximation of the Stokes Problem , 1988 .

[9]  R. Temam.  Navier-Stokes Equations , 1977 .

[10]  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.

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

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

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

[14]  F. Musy,et al.  A Fast Solver for the Stokes Equations Using Multigrid with a UZAWA Smoother , 1985 .

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

[16]  R. Temam,et al.  Navier-Stokes equations: theory and numerical analysis: R. Teman North-Holland, Amsterdam and New York. 1977. 454 pp. US $45.00 , 1978 .

[17]  R. Glowinski,et al.  Numerical methods for the navier-stokes equations. Applications to the simulation of compressible and incompressible viscous flows , 1987 .

[18]  P. Fischer,et al.  Parallel spectral element solution of the Stokes problem , 1991 .

[19]  Yvon Maday,et al.  Relèvement polynômial de traces et applications , 1990 .

[20]  Yvon Maday,et al.  Spectral approximation of the periodic-nonperiodic Navier-Stokes equations , 1987 .

[21]  A. Patera.  A spectral element method for fluid dynamics: Laminar flow in a channel expansion , 1984 .

[22]  Rüdiger Verfürth.  A Preconditioned Conjugate Residual Algorithm for the Stokes Problem , 1985 .

[23]  R. Glowinski,et al.  Numerical Methods for Nonlinear Variational Problems , 1985 .