## Abstract

We first summarize the general concept of our verification method of solutions for elliptic equations. Next, as an application of our method, a survey and future works on the numerical verification method of solutions for heat convection problems known as Rayleigh-Bénard problem are described. We will give a method to verify the existence of bifurcating solutions of the two-dimensional problem and the bifurcation point itself. Finally, an extension to the three-dimensional case and future works will be described.

### Similar content being viewed by others

## References

H. Bénard, Les tourbillons cellulaires dans une nappe liquide. Revue Gén. Sci. Pure Appl.,

**11**(1900), 1261–1271, 1309–1328.S. Chandrasekhar, Hydrodynamic and Hydromagnetic Stability. Oxford University Press, 1961.

J.H. Curry, Bounded solutions of finite dimensional approximations to the Boussinesq equations. SIAM J. Math. Anal.,

**10**(1979), 71–79.A.V. Getling, Rayleigh-Bénard Convection: Structures and Dynamics. Advanced Series in Nonlinear Dynamics,

**11**, World Scientific, 1998.V.I. Iudovich, On the origin of convection. J. Appl. Math. Mech.,

**30**(1966), 1193–1199.D.D. Joseph, On the stability of the Boussinesq equations. Arch. Rational Mech. Anal.,

**20**(1965), 59–71.Y. Kagei and W. von Wahl, The Eckhaus criterion for convection roll solutions of the Oberbeck-Boussinesq equations. Int. J. Non-linear Mechanics.,

**32**(1997), 563–620.T. Kawanago, A symmetry-breaking bifurcation theorem and some related theorems applicable to maps having unbounded derivatives. Japan J. Indust. Appl. Math,

**21**(2004), 57–74.F. Kikuchi and X. Xuefeng, Determination of the Babuska-Aziz constant for the linear triangular finite element. Japan J. Ind. Appl. Math.,

**23**(2006), 75–82.M.-N. Kim, M.T. Nakao, Y. Watanabe and T. Nishida, A numerical verification method of bifurcating solutions for 3-dimensional Rayleigh-Bénard problems. Numer. Math.,

**111**(2009), 389–406.O. Knüppel, PROFIL/BIAS—A fast interval library. Computing,

**53**(1994), 277–287, http://www.ti3.tu-harburg.de/Software/PROFILEnglisch.html.R. Krishnamurti, Some further studies on the transition to turbulent convection. J. Fluid Mech.,

**60**(1973), 285–303.K. Nagatou, N. Yamamoto and M.T. Nakao, An approach to the numerical verification of solutions for nonlinear elliptic problems with local uniqueness. Numer. Funct. Anal. Optim.,

**20**(1999), 543–565.K. Nagatou, K. Hashimoto and M.T. Nakao, Numerical verification of stationary solutions for Navier-Stokes problems. J. Comput. Appl. Math.,

**199**(2007), 424–431.M.T. Nakao, A numerical approach to the proof of existence of solutions for elliptic problems. Japan J. Appl. Math.,

**5**(1988), 313–332.M.T. Nakao, N. Yamamoto and S. Kimura, On best constant in the optimal error stimates for the

*H*^{1}_{0}-projection into piecewise polynomial spaces. Journal of Approximation. Theory,**93**, (1998), 491–500.M.T. Nakao, Numerical verification methods for solutions of ordinary and partial differential equations. Numer. Funct. Anal. Optim.,

**22**(2001), 321–356.M.T. Nakao, K. Hashimoto and Y. Watanabe, A numerical method to verify the invertibility of linear elliptic operators with applications to nonlinear problems. Computing

**75**(2005), 1–14.M.T. Nakao, Y. Watanabe, N. Yamamoto and T. Nishida, Some computer assisted proofs for solutions of the heat convection problems. Reliable Computing,

**9**(2003), 359–372.M.T. Nakao and Y. Watanabe, An efficient approach to the numerical verification for solutions of elliptic differential equations. Numer. Algor.,

**37**(2004), 311–323.T. Nishida, T. Ikeda and H. Yoshihara, Pattern formation of heat convection problems. Proceedings of the International Symposium on Mathematical Modeling and Numerical Simulation in Continuum Mechanics, T. Miyoshi et al. (eds.), Lecture Notes in Computational Science and Engineering,

**19**, Springer-Verlag, 2002, 155–167.M. Plum, Explicit

*H*_{2}-estimates, and pointwise bounds for solutions of second-order elliptic boundary value problems. J. Math. Anal. Appl.,**165**(1992), 36–61.P.H. Rabinowitz, Existence and nonuniqueness of rectangular solutions of the Bénard problem. Arch. Rational Mech. Anal.,

**29**(1968), 32–57.J.W.S. Rayleigh, On convection currents in a horizontal layer of fluid, when the higher temperature is on the under side. Phil. Mag.,

**32**(1916), 529–546; Sci. Papers,**6**, 432–446.S.M. Rump, On the solution of interval linear systems. Computing,

**47**(1992), 337–353.S.M. Rump, A note on epsilon-inflation. Reliable Computing,

**4**(1998), 371–375.Y. Watanabe, N. Yamamoto, M.T. Nakao and T. Nishida, A numerical verification of nontrivial solutions for the heat convection problem. J. Math. Fluid Mech.,

**6**(2004), 1–20.Y. Watanabe, A computer-assisted proof for the Kolmogorov flows of incompressible viscous fluid. J. Comput. Appl. Math.,

**223**(2009), 953–966.N. Yamamoto, A numerical verification method for solutions of boundary value problems with local uniqueness by Banach’s fixed point theorem. SIAM J. Numer. Anal.,

**35**(1998), 2004–2013.

## Author information

### Authors and Affiliations

### Corresponding author

## About this article

### Cite this article

Watanabe, Y., Nakao, M.T. Numerical verification method of solutions for elliptic equations and its application to the Rayleigh-Bénard problem.
*Japan J. Indust. Appl. Math.* **26**, 443–463 (2009). https://doi.org/10.1007/BF03186543

Received:

Revised:

Issue Date:

DOI: https://doi.org/10.1007/BF03186543