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On the convergence of spectral approximations for the heat convection equations

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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.

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References

  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)

  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)

    Article  MathSciNet  MATH  Google Scholar 

  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)

  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)

    MathSciNet  Google Scholar 

  5. Cattabriga, L.: Su un problema al contorno relativo al sistema di equazioni di Stokes. Rend. Sem. Mat. Univ. Padova 31, 308–340 (1961)

    MathSciNet  MATH  Google Scholar 

  6. Fujita, H., Kato, T.: On the Navier-Stokes initial value problem I. Arch. Ration. Mech. Anal. 16, 269–315 (1964)

    Article  MathSciNet  MATH  Google Scholar 

  7. Hishida, T.: Existence and regularizing properties of solutions for the nonstationary convection problem. Funkcialy Ekvaciy 34, 449–474 (1991)

    MathSciNet  MATH  Google Scholar 

  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)

    MathSciNet  MATH  Google Scholar 

  9. Heywood, J.G.: The Navier-Stokes equations: on the existence, regularity and decay of solutions. Indiana Univ. Math. J. 9, 639–681 (1980)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  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)

  14. Jörgens, K., Rellich, F.: Eigenwerttheorie gewöhnlicher Differentialgleichungen. Springer, Berlin (1976)

    Book  MATH  Google Scholar 

  15. Joseph, D.D.: Stability of fluid motion. Springer, Berlin (1976)

    Google Scholar 

  16. Ilyin, A.A.: On the spectrum of the Stokes operator. Funct. Anal. Appl. 43, 14–25 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    MathSciNet  Google Scholar 

  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. Ladyzhenskaya, O.A.: The mathematical theory of viscous incompressible fluid. Gordon and Breach, New York (1969)

    MATH  Google Scholar 

  20. Lions, J.L.: Quelques méthodes de résolution des problèmes aux limites nonlinéaires. Dunod, Paris (1969)

    MATH  Google Scholar 

  21. Mikhailov, V.: Équations aux Dérivées Partielles. Mir, Moscow (1980)

    Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    MathSciNet  MATH  Google Scholar 

  24. Morimoto, H.: Nonstationary Boussinesq equations. J. Fac. Sci. Univ. Tokyo Sect. IA Math. 39, 61–75 (1992)

    MathSciNet  MATH  Google Scholar 

  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)

    MATH  Google Scholar 

  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)

  27. Rautmann, R.: On optimum regularity of Navier-Stokes solutions at time \(t=0\). Math. Z. 184, 141–149 (1983)

    Article  MathSciNet  MATH  Google Scholar 

  28. Rautmann, R.: A semigroup approach to error estimates for nonstationary Navier-Stokes approximations. Methoden Verfahren Math. Physik 27, 63–77 (1983)

    MathSciNet  MATH  Google Scholar 

  29. Rautmann, R.: \(H^2\)-convergence of Rothe’s scheme to the Navier-Stokes equations. Nonlinear Anal. 24, 1081–1102 (1995)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Article  MATH  Google Scholar 

  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)

    MathSciNet  MATH  Google Scholar 

  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)

    MathSciNet  MATH  Google Scholar 

  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)

    MathSciNet  MATH  Google Scholar 

  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)

    MathSciNet  MATH  Google Scholar 

  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)

    MathSciNet  MATH  Google Scholar 

  36. Rummler, B.: The eigenfunctions of the Stokes operator in special domains. I. ZAMM 77, 619–627 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  37. Rummler, B.: The eigenfunctions of the Stokes operator in special domains. II. ZAMM 77, 669–675 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  38. Salvi, R.: Error estimates for spectral Galerkin approximations of the solutions of Navier-Stokes type equations. Glasgow Math. J. 31, 199–211 (1989)

    Article  MathSciNet  MATH  Google Scholar 

  39. Shinbroth, M., Kotorynski, W.P.: The initial value problem for a viscous heat-conduting fluid. J. Math. Anal. Appl. 45, 1–22 (1974)

    Article  MathSciNet  Google Scholar 

  40. Sohr, H.: The Navier-Stokes Equations, an elementary functional analytic approach. Birkauser, Bassel (2001)

    MATH  Google Scholar 

  41. Temam, R.: Navier-Stokes equations, theory and numerical analysis, North-Holland, 2nd revised edn. Amsterdam (1979)

  42. Temam, R.: Behaviour at time \(t=0\) of the solutions of semi-linear evolution equations. J. Differ. Equations 43, 73–92 (1982)

    Article  MathSciNet  MATH  Google Scholar 

  43. Vinogradova, P., Zarubin, A.: A study of Galerkin method for the heat convection equations. Appl. Math. Comput. 218, 520–531 (2011)

    Article  MathSciNet  MATH  Google Scholar 

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Authors and Affiliations

  1. Dpto. de Ecuaciones Diferenciales y Análisis Numérico, Universidad de Sevilla, Seville, Spain

    B. Climent-Ezquerra

  2. Dpto. de Matemática, Universidad de Atacama, Copiapó, Chile

    M. Poblete-Cantellano

  3. Instituto de Alta Investigación, Universidad de Tarapacá, Casilla 7D, Arica, Chile

    M. A. Rojas-Medar

Authors
  1. B. Climent-Ezquerra
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  2. M. Poblete-Cantellano
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  3. M. A. Rojas-Medar
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Correspondence to B. Climent-Ezquerra.

Additional information

MTM2012-12927, Spain. Universidad Atacama, project 221169 DIUDA 8/3. Fondecyt-Chile, Grant 1120260.

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Climent-Ezquerra, B., Poblete-Cantellano, M. & Rojas-Medar, M.A. On the convergence of spectral approximations for the heat convection equations. Rev Mat Complut 29, 405–422 (2016). https://doi.org/10.1007/s13163-016-0189-y

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  • DOI: https://doi.org/10.1007/s13163-016-0189-y

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