Long-term dissipativity of time-stepping algorithms for an abstract evolution equation with applications to the incompressible MHD and Navier-Stokes equations☆

[1]  P. Gresho.  Some current CFD issues relevant to the incompressible Navier-Stokes equations , 1991 .

[2]  I. Simonov.  Small steady perturbations behind the front of an oblique shock wave , 1979 .

[3]  H. Goldstein,et al.  Classical Mechanics , 1951, Mathematical Gazette.

[4]  B. Mercier,et al.  Eigenvalue approximation by mixed and hybrid methods , 1981 .

[5]  V. Maz'ya,et al.  Elliptic Boundary Value Problems , 1984 .

[6]  John C. Butcher,et al.  A stability property of implicit Runge-Kutta methods , 1975 .

[7]  P. Raviart,et al.  Conforming and nonconforming finite element methods for solving the stationary Stokes equations I , 1973 .

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

[9]  Rolf Rannacher,et al.  Finite element approximation of the nonstationary Navier-Stokes problem, part II: Stability of solutions and error estimates uniform in time , 1986 .

[10]  R. Temam.  Une méthode d'approximation de la solution des équations de Navier-Stokes , 1968 .

[11]  M. Powell.  A method for nonlinear constraints in minimization problems , 1969 .

[12]  F. B. Ellerby,et al.  Numerical solutions of partial differential equations by the finite element method , by C. Johnson. Pp 278. £40 (hardback), £15 (paperback). 1988. ISBN 0-521-34514-6, 34758-0 (Cambridge University Press) , 1989, The Mathematical Gazette.

[13]  R. Glowinski,et al.  Augmented Lagrangian and Operator-Splitting Methods in Nonlinear Mechanics , 1987 .

[14]  I. Babuska.  The finite element method with Lagrangian multipliers , 1973 .

[15]  J. C. Simo,et al.  Conserving algorithms for the dynamics of Hamiltonian systems on lie groups , 1994 .

[16]  Jean Leray,et al.  Essai sur les mouvements plans d'un fluide visqueux que limitent des parois. , 1934 .

[17]  Edriss S. Titi,et al.  Dissipativity of numerical schemes , 1991 .

[18]  J. A. Shercliff,et al.  A Textbook of Magnetohydrodynamics , 1966 .

[19]  C. Scovel.  Symplectic Numerical Integration of Hamiltonian Systems , 1991 .

[20]  J. Marsden,et al.  Groups of diffeomorphisms and the motion of an incompressible fluid , 1970 .

[21]  A. Chorin.  Numerical solution of the Navier-Stokes equations , 1968 .

[22]  M. Gunzburger,et al.  On the existence, uniqueness, and finite element approximation of solutions of the equations of stationary, incompressible magnetohydrodynamics , 1991 .

[23]  M. Fortin,et al.  Augmented Lagrangian methods : applications to the numerical solution of boundary-value problems , 1983 .

[24]  Jean Leray,et al.  Étude de diverses équations intégrales non linéaires et de quelques problèmes que pose l'Hydrodynamique. , 1933 .

[25]  J. C. Simo,et al.  Unconditional stability and long-term behavior of transient algorithms for the incompressible Navier-Stokes and Euler equations , 1994 .

[26]  Jack K. Hale,et al.  Upper semicontinuity of attractors for approximations of semigroups and partial differential equations , 1988 .

[27]  Roger Temam,et al.  Some mathematical questions related to the MHD equations , 1983 .

[28]  J. Hale.  Asymptotic Behavior of Dissipative Systems , 1988 .

[29]  Vivette Girault,et al.  Finite Element Methods for Navier-Stokes Equations - Theory and Algorithms , 1986, Springer Series in Computational Mathematics.

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

[31]  J. Marsden,et al.  A mathematical introduction to fluid mechanics , 1979 .

[32]  Andrew M. Stuart,et al.  Runge-Kutta methods for dissipative and gradient dynamical systems , 1994 .

[33]  V. Arnold.  Mathematical Methods of Classical Mechanics , 1974 .

[34]  J. Z. Zhu,et al.  The finite element method , 1977 .

[35]  M. Vishik,et al.  Attractors of Evolution Equations , 1992 .

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

[37]  R. Temam.  Infinite Dimensional Dynamical Systems in Mechanics and Physics Springer Verlag , 1993 .

[38]  C. Foiaș,et al.  Sur le comportement global des solutions non-stationnaires des équations de Navier-Stokes en dimension $2$ , 1967 .

[39]  G. Strang,et al.  An Analysis of the Finite Element Method , 1974 .

[40]  R. Kohn,et al.  Partial regularity of suitable weak solutions of the navier‐stokes equations , 1982 .

[41]  Jie Shen,et al.  Convergence of approximate attractors for a fully discrete system for reaction-diffusion equations , 1989 .

[42]  J. C. Simo,et al.  Quasi-incompressible finite elasticity in principal stretches. Continuum basis and numerical algorithms , 1991 .

[43]  R. Temam.  Navier-Stokes Equations and Nonlinear Functional Analysis , 1987 .

[44]  R. A. Silverman,et al.  The Mathematical Theory of Viscous Incompressible Flow , 1972 .

[45]  Gérard A. Maugin,et al.  Electrodynamics Of Continua , 1990 .

[46]  M. Hestenes.  Multiplier and gradient methods , 1969 .

[47]  J. C. Simo,et al.  Unconditionally stable algorithms for rigid body dynamics that exactly preserve energy and momentum , 1991 .

[48]  M. Fortin,et al.  A generalization of Uzawa's algorithm for the solution of the Navier-Stokes equations , 1985 .

[49]  Jerrold E. Marsden,et al.  Nonlinear stability of fluid and plasma equilibria , 1985 .

[50]  E. M. Lifshitz,et al.  Electrodynamics of continuous media , 1961 .

[51]  V. Arnold.  Sur la géométrie différentielle des groupes de Lie de dimension infinie et ses applications à l'hydrodynamique des fluides parfaits , 1966 .

[52]  R. Rannacher,et al.  Finite element approximation of the nonstationary Navier-Stokes problem. I : Regularity of solutions and second-order error estimates for spatial discretization , 1982 .

[53]  W. Hughes,et al.  Electromagnetodynamics of fluids , 1966 .

[54]  J. L. Lions,et al.  Inéquations en thermoélasticité et magnétohydrodynamique , 1972 .

[55]  R. Temam,et al.  Asymptotic numerical analysis for the Navier-Stokes equations, 1 , 1982 .

[56]  R. Temam,et al.  On the Large Time Galerkin Approximation of the Navier–Stokes Equations , 1984 .

[57]  D. A. Dunnett.  Classical Electrodynamics , 2020, Nature.

[58]  J. Marsden,et al.  Lie-Poisson Hamilton-Jacobi theory and Lie-Poisson integrators , 1988 .

[59]  Daniel D. Joseph,et al.  Nonlinear dynamics and turbulence , 1983 .

[60]  Semi-discretized nonlinear evolution equations as discrete dynamical systems and error analysis , 1990 .

[61]  J. C. Simo,et al.  The discrete energy-momentum method. Conserving algorithms for nonlinear elastodynamics , 1992 .

[62]  Michel Fortin,et al.  Mixed and Hybrid Finite Element Methods , 2011, Springer Series in Computational Mathematics.