zbMATH — the first resource for mathematics

Geometry Search for the term Geometry in any field. Queries are case-independent.
Funct* Wildcard queries are specified by * (e.g. functions, functorial, etc.). Otherwise the search is exact.
"Topological group" Phrases (multi-words) should be set in "straight quotation marks".
au: Bourbaki & ti: Algebra Search for author and title. The and-operator & is default and can be omitted.
Chebyshev | Tschebyscheff The or-operator | allows to search for Chebyshev or Tschebyscheff.
"Quasi* map*" py: 1989 The resulting documents have publication year 1989.
so: Eur* J* Mat* Soc* cc: 14 Search for publications in a particular source with a Mathematics Subject Classification code (cc) in 14.
"Partial diff* eq*" ! elliptic The not-operator ! eliminates all results containing the word elliptic.
dt: b & au: Hilbert The document type is set to books; alternatively: j for journal articles, a for book articles.
py: 2000-2015 cc: (94A | 11T) Number ranges are accepted. Terms can be grouped within (parentheses).
la: chinese Find documents in a given language. ISO 639-1 language codes can also be used.

a & b logic and
a | b logic or
!ab logic not
abc* right wildcard
"ab c" phrase
(ab c) parentheses
any anywhere an internal document identifier
au author, editor ai internal author identifier
ti title la language
so source ab review, abstract
py publication year rv reviewer
cc MSC code ut uncontrolled term
dt document type (j: journal article; b: book; a: book article)
Numerical verification method of solutions for elliptic equations and its application to the Rayleigh-Bénard problem. (English) Zbl 1184.65106
Summary: 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.

65N30Finite elements, Rayleigh-Ritz and Galerkin methods, finite methods (BVP of PDE)
65C20Models (numerical methods)
35B32Bifurcation (PDE)
35J66Nonlinear boundary value problems for nonlinear elliptic equations
65N15Error bounds (BVP of PDE)
Full Text: DOI Euclid
[1] H. Bénard, Les tourbillons cellulaires dans une nappe liquide. Revue Gén. Sci. Pure Appl.,11 (1900), 1261--1271, 1309--1328.
[2] S. Chandrasekhar, Hydrodynamic and Hydromagnetic Stability. Oxford University Press, 1961.
[3] J.H. Curry, Bounded solutions of finite dimensional approximations to the Boussinesq equations. SIAM J. Math. Anal.,10 (1979), 71--79. · Zbl 0395.35073 · doi:10.1137/0510008
[4] A.V. Getling, Rayleigh-Bénard Convection: Structures and Dynamics. Advanced Series in Nonlinear Dynamics,11, World Scientific, 1998.
[5] V.I. Iudovich, On the origin of convection. J. Appl. Math. Mech.,30 (1966), 1193--1199. · doi:10.1016/0021-8928(66)90081-5
[6] D.D. Joseph, On the stability of the Boussinesq equations. Arch. Rational Mech. Anal.,20 (1965), 59--71. · Zbl 0136.23402 · doi:10.1007/BF00250190
[7] 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. · Zbl 0891.76035 · doi:10.1016/S0020-7462(97)88306-0
[8] 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. · Zbl 1054.37030 · doi:10.1007/BF03167432
[9] 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. · Zbl 1098.65107 · doi:10.1007/BF03167499
[10] 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. · Zbl 1155.76024 · doi:10.1007/s00211-008-0191-5
[11] O. Knüppel, PROFIL/BIAS--A fast interval library. Computing,53 (1994), 277--287, http://www.ti3.tu-harburg.de/Software/PROFILEnglisch.html. · Zbl 0808.65055 · doi:10.1007/BF02307379
[12] R. Krishnamurti, Some further studies on the transition to turbulent convection. J. Fluid Mech.,60 (1973), 285--303. · doi:10.1017/S0022112073000170
[13] 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. · Zbl 0938.65137 · doi:10.1080/01630569908816910
[14] 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. · Zbl 1106.76057 · doi:10.1016/j.cam.2005.09.031
[15] M.T. Nakao, A numerical approach to the proof of existence of solutions for elliptic problems. Japan J. Appl. Math.,5 (1988), 313--332. · Zbl 0694.35051 · doi:10.1007/BF03167877
[16] M.T. Nakao, N. Yamamoto and S. Kimura, On best constant in the optimal error stimates for theH 0 1 -projection into piecewise polynomial spaces. Journal of Approximation. Theory,93, (1998), 491--500. · Zbl 0907.65100
[17] M.T. Nakao, Numerical verification methods for solutions of ordinary and partial differential equations. Numer. Funct. Anal. Optim.,22 (2001), 321--356. · Zbl 1106.65315 · doi:10.1081/NFA-100105107
[18] M.T. Nakao, K. Hashimoto and Y. Watanabe, A numerical method to verify the invertibility of linear elliptic operators with applications to nonlinear problems. Computing75 (2005), 1--14. · Zbl 1151.35337 · doi:10.1007/s00607-004-0111-1
[19] 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. · Zbl 1126.35352 · doi:10.1023/A:1025179130399
[20] 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. · Zbl 1114.65152 · doi:10.1023/B:NUMA.0000049477.75366.94
[21] 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. · Zbl 1001.76033
[22] M. Plum, ExplicitH 2-estimates, and pointwise bounds for solutions of second-order elliptic boundary value problems. J. Math. Anal. Appl.,165 (1992), 36--61. · Zbl 0780.35028 · doi:10.1016/0022-247X(92)90067-N
[23] P.H. Rabinowitz, Existence and nonuniqueness of rectangular solutions of the Bénard problem. Arch. Rational Mech. Anal.,29 (1968), 32--57. · Zbl 0164.28704 · doi:10.1007/BF00256457
[24] 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. · Zbl 46.1249.04
[25] S.M. Rump, On the solution of interval linear systems. Computing,47 (1992), 337--353. · Zbl 0753.65030 · doi:10.1007/BF02320201
[26] S.M. Rump, A note on epsilon-inflation. Reliable Computing,4 (1998), 371--375. · Zbl 0920.65031 · doi:10.1023/A:1024419816707
[27] 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. · Zbl 1062.35092 · doi:10.1007/s00021-003-0077-3
[28] Y. Watanabe, A computer-assisted proof for the Kolmogorov flows of incompressible viscous fluid. J. Comput. Appl. Math.,223 (2009), 953--966. · Zbl 1153.76022 · doi:10.1016/j.cam.2008.03.034
[29] 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. · Zbl 0972.65084 · doi:10.1137/S0036142996304498