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Revista Matemática Complutense

, Volume 29, Issue 2, pp 405–422 | Cite as

On the convergence of spectral approximations for the heat convection equations

  • B. Climent-EzquerraEmail author
  • M. Poblete-Cantellano
  • M. A. Rojas-Medar
Article
  • 120 Downloads

Abstract

In this paper, we focus on the convergence rate of solutions of spectral Galerkin approximations for the heat convection equations on a bounded domain. Estimates in \(H^2\)-norm for velocity and temperature without compatibility conditions are obtained. Moreover, we give rates of convergence for the velocity and temperature derivatives in \(L^ 2\)-norm.

Keywords

Spectral Galerkin approximations Convergence rate  Boussinesq equations Navier-Stokes type equations Heat convection equations 

Mathematics Subject Classification

65M60 76D05 65M15 76R99 

References

  1. 1.
    Amrouche, C., Girault, V.: On the existence and regularity of the solution of Stokes problem in arbitrary dimension. Proc. Jpn Acad., 67(Ser. A), 171–175 (1991)Google Scholar
  2. 2.
    Bause, M.: On optimal convergence rates for higher-order Navier-Stokes approximations. I. Error estimates for the spatial discretization. IMA J. Numer. Anal. 25, 812–841 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Cannon, J.R., DiBenedetto, E.: The initial value problem for the Boussinesq equations with data in \(L^p,\) Approximation Methods for Navier-Stokes Equations. Springer Lect. Notes 771, 129–144 (1979)Google Scholar
  4. 4.
    Cabrales, R.C., Poblete-Cantellano, M., Rojas-Medar, M.A.: Ecuaciones de Boussinesq: estimaciones uniformes en el tiempo de las aproximaciones de Galerkin espectrales. Revista Integración 27(1), 37–57 (2009)MathSciNetGoogle Scholar
  5. 5.
    Cattabriga, L.: Su un problema al contorno relativo al sistema di equazioni di Stokes. Rend. Sem. Mat. Univ. Padova 31, 308–340 (1961)MathSciNetzbMATHGoogle Scholar
  6. 6.
    Fujita, H., Kato, T.: On the Navier-Stokes initial value problem I. Arch. Ration. Mech. Anal. 16, 269–315 (1964)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Hishida, T.: Existence and regularizing properties of solutions for the nonstationary convection problem. Funkcialy Ekvaciy 34, 449–474 (1991)MathSciNetzbMATHGoogle Scholar
  8. 8.
    Heywood, J.G.: An error estimate uniform in time for spectral Galerkin approximations of the Navier-Stokes problem. P. J. Math. 96, 33–345 (1982)MathSciNetzbMATHGoogle Scholar
  9. 9.
    Heywood, J.G.: The Navier-Stokes equations: on the existence, regularity and decay of solutions. Indiana Univ. Math. J. 9, 639–681 (1980)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Heywood, J.G., Rannacher, R.: Finite element approximation of the nonstationary Navier-Stokes problem I: regularity of solutions and second order error estimates for spatial discretization. SIAM J. Numer. Anal. 19, 275–311 (1982)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Heywood, J.G., Rannacher, R.: Finite element approximation of the nonstationary Navier-Stokes problem II: stability of solutions and error estimates uniform in time. SIAM J. Numer. Anal. 23, 750–777 (1986)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Heywood, J.G., Rannacher, R.: Finite element approximation of the nonstationary Navier-Stokes problem III: smoothing property and higher order error estimates for spatial discretizations. SIAM J. Numer. Anal. 25, 489–512 (1988)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Heywood, J.G., Rannacher, R.: Finite element approximation of the nonstationary Navier-Stokes problem IV: error analysis for second-order time discretization. SIAM J. Numer. Anal. 27, 353–384 (1990)Google Scholar
  14. 14.
    Jörgens, K., Rellich, F.: Eigenwerttheorie gewöhnlicher Differentialgleichungen. Springer, Berlin (1976)CrossRefzbMATHGoogle Scholar
  15. 15.
    Joseph, D.D.: Stability of fluid motion. Springer, Berlin (1976)Google Scholar
  16. 16.
    Ilyin, A.A.: On the spectrum of the Stokes operator. Funct. Anal. Appl. 43, 14–25 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Kiselev, A.A., Ladyzhenskaya, O.A.: On the existence and uniqueness of the solution of the nonstationary problem for a viscous incompressible fluid. Izv. Akad. Nauk. SSSR, Ser. Mat. 21, 655–680 (1957)MathSciNetGoogle Scholar
  18. 18.
    Korenev, N.K.: On some problems of convection in a viscous incompressible fluid. Vestnik Leningrand Univ. math. 4, 125–137 (1977)Google Scholar
  19. 19.
    Ladyzhenskaya, O.A.: The mathematical theory of viscous incompressible fluid. Gordon and Breach, New York (1969)zbMATHGoogle Scholar
  20. 20.
    Lions, J.L.: Quelques méthodes de résolution des problèmes aux limites nonlinéaires. Dunod, Paris (1969)zbMATHGoogle Scholar
  21. 21.
    Mikhailov, V.: Équations aux Dérivées Partielles. Mir, Moscow (1980)Google Scholar
  22. 22.
    Moretti, A.C., Rojas-Medar, M.A., Rojas-Medar, M.D.: The equations of a viscous incompressible chemically active fluid: existence and uniqueness of strong solutions in an unbounded domain. Comput. Math. Appl. 44(3–4), 287–299 (2002)MathSciNetCrossRefzbMATHGoogle Scholar
  23. 23.
    Morimoto, H.: On the existence of weak solutions of the equation of natural convection. J. Fac. Sci. Univ. Tokio, Sect. IA 36, 87–102 (1989)MathSciNetzbMATHGoogle Scholar
  24. 24.
    Morimoto, H.: Nonstationary Boussinesq equations. J. Fac. Sci. Univ. Tokyo Sect. IA Math. 39, 61–75 (1992)MathSciNetzbMATHGoogle Scholar
  25. 25.
    Okamoto, H.: On the semidiscrete finite element approximations for the nonstationary Navier-Stokes equations. J. Fac. Sci. Univ. Tokyo IA 29, 613–652 (1982)zbMATHGoogle Scholar
  26. 26.
    Rautmann, R.: On the convergence rate of nonstationary Navier-Stokes approximations, Proc. IUTAM Symp. 1979, Approx. Methods for Navier-Stokes problem. In: Rautmann, R. (ed.) Lect. Notes in Math, vol. 771, pp. 425–449. Springer, New York (1980)Google Scholar
  27. 27.
    Rautmann, R.: On optimum regularity of Navier-Stokes solutions at time \(t=0\). Math. Z. 184, 141–149 (1983)MathSciNetCrossRefzbMATHGoogle Scholar
  28. 28.
    Rautmann, R.: A semigroup approach to error estimates for nonstationary Navier-Stokes approximations. Methoden Verfahren Math. Physik 27, 63–77 (1983)MathSciNetzbMATHGoogle Scholar
  29. 29.
    Rautmann, R.: \(H^2\)-convergence of Rothe’s scheme to the Navier-Stokes equations. Nonlinear Anal. 24, 1081–1102 (1995)MathSciNetCrossRefzbMATHGoogle Scholar
  30. 30.
    Rodríguez-Bellido, M.A., Rojas-Medar, M.A., Villamizar-Roa, E.J.: Periodic solutions in unbounded domains to the Boussinesq equations. Acta Math. Sinica. Engl. Ser. 26(5), 837–862 (2010)CrossRefzbMATHGoogle Scholar
  31. 31.
    Rojas-Medar, M.A., Boldrini, J.L.: Spectral Galerkin approximations for the Navier-Stokes equations : uniform in time error estimates. Rev. Mat. Apl. 14, 63–74 (1993)MathSciNetzbMATHGoogle Scholar
  32. 32.
    Rojas-Medar, M.A., Lorca, S.A.: The equations of a viscous incompressible chemical active fluid I: uniqueness and existence of the local solutions. Rev. Mat. Apl. 16, 57–80 (1995)MathSciNetzbMATHGoogle Scholar
  33. 33.
    Rojas-Medar, M.A., Lorca, S.A.: The equations of a viscous incompressible chemical active fluid I: regularity of solutions. Rev. Mat. Apl. 16, 81–95 (1995)MathSciNetzbMATHGoogle Scholar
  34. 34.
    Rojas-Medar, M.A., Lorca, S.A.: Global strong solution of the equations for the motion of a chemical active fluid. Mat. Contemp. 8, 319–335 (1995)MathSciNetzbMATHGoogle Scholar
  35. 35.
    Rojas-Medar, M.A., Lorca, S.A.: An error estimate uniform in time for spectral Galerkin approximations for the equations for the motion of a chemical active fluid. Rev. Matemática de la Universidad Complutense de Madrid 8, 431–458 (1995)MathSciNetzbMATHGoogle Scholar
  36. 36.
    Rummler, B.: The eigenfunctions of the Stokes operator in special domains. I. ZAMM 77, 619–627 (1997)MathSciNetCrossRefzbMATHGoogle Scholar
  37. 37.
    Rummler, B.: The eigenfunctions of the Stokes operator in special domains. II. ZAMM 77, 669–675 (1997)MathSciNetCrossRefzbMATHGoogle Scholar
  38. 38.
    Salvi, R.: Error estimates for spectral Galerkin approximations of the solutions of Navier-Stokes type equations. Glasgow Math. J. 31, 199–211 (1989)MathSciNetCrossRefzbMATHGoogle Scholar
  39. 39.
    Shinbroth, M., Kotorynski, W.P.: The initial value problem for a viscous heat-conduting fluid. J. Math. Anal. Appl. 45, 1–22 (1974)MathSciNetCrossRefGoogle Scholar
  40. 40.
    Sohr, H.: The Navier-Stokes Equations, an elementary functional analytic approach. Birkauser, Bassel (2001)zbMATHGoogle Scholar
  41. 41.
    Temam, R.: Navier-Stokes equations, theory and numerical analysis, North-Holland, 2nd revised edn. Amsterdam (1979)Google Scholar
  42. 42.
    Temam, R.: Behaviour at time \(t=0\) of the solutions of semi-linear evolution equations. J. Differ. Equations 43, 73–92 (1982)MathSciNetCrossRefzbMATHGoogle Scholar
  43. 43.
    Vinogradova, P., Zarubin, A.: A study of Galerkin method for the heat convection equations. Appl. Math. Comput. 218, 520–531 (2011)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Universidad Complutense de Madrid 2016

Authors and Affiliations

  • B. Climent-Ezquerra
    • 1
    Email author
  • M. Poblete-Cantellano
    • 2
  • M. A. Rojas-Medar
    • 3
  1. 1.Dpto. de Ecuaciones Diferenciales y Análisis NuméricoUniversidad de SevillaSevilleSpain
  2. 2.Dpto. de MatemáticaUniversidad de AtacamaCopiapóChile
  3. 3.Instituto de Alta InvestigaciónUniversidad de TarapacáAricaChile

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