Abstract
The verification algorithm FS-Int, which was introduced in the previous chapter, succeeds in enclosing solutions for several realistic problems. However, to obtain the fixed point of F in (1.9), it is necessary that the map F is retractive in some neighborhood of the fixed point to be verified. In order to apply our verification method to more general problems, we introduce some Newton-type verification algorithms.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Notes
- 1.
Throughout Part I, we use symbols of differential operators like f′[v], \(N_h^{\prime }[w]\), T′[w], etc. They differ from other parts.
References
Lions, P.-L.: On the existence of positive solutions of semilinear elliptic equations. SIAM Rev. 24(4), 441–467 (1982)
Nakao, M.T.: Solving nonlinear elliptic problems with result verification using an H −1 type residual iteration. In: Validation Numerics: Theory and Applications. Volume 9 of Computing Supplementum, pp. 161–173. Springer, Vienna (1993)
Nakao, M.T.: A computational verification method of existence of solutions for nonlinear elliptic equations. In: Recent Topics in Nonlinear PDE, IV, Kyoto, 1988. Volume 160 of North-Holland Mathematics Studies, pp. 101–120. North-Holland, Amsterdam (1989)
Nakao, M.T.: A numerical approach to the proof of existence of solutions for elliptic problems. II. Japan J. Appl. Math. 7(3), 477–488 (1990)
Nakao, M.T.: Computable error estimates for FEM and numerical verification of solutions for nonlinear PDEs. In: Computational and Applied Mathematics, I (Dublin, 1991), pp. 357–366. North-Holland, Amsterdam (1992)
Nakao, M.T.: A numerical verification method for the existence of weak solutions for nonlinear boundary value problems. J. Math. Anal. Appl. 164(2), 489–507 (1992)
Nakao, M.T.: Numerical verification methods for solutions of ordinary and partial differential equations. Numer. Funct. Anal. Optim. 22(3–4), 321–356 (2001). International Workshops on Numerical Methods and Verification of Solutions, and on Numerical Function Analysis, Ehime/Shimane, 1999
Nakao, M.T., Watanabe, Y.: An efficient approach to the numerical verification for solutions of elliptic differential equations. Numer. Algorithms 37(1–4), 311–323 (2004)
Nakao, M.T., Watanabe, Y.: Numerical verification methods for solutions of semilinear elliptic boundary value problems. Nonlinear Theory and Its Applications, IEICE 2(1), 2–31 (2011)
Nakao, M.T., Watanabe, Y.: Self-Validating Numerical Computations by Learning from Examples: Theory and Implementation. Volume 85 of The Library for Senior & Graduate Courses. Saiensu-sha (in Japanese), Tokyo (2011)
Nakao, M.T., Yamamoto, N.: Numerical verifications of solutions for elliptic equations with strong nonlinearity. Numer. Funct. Anal. Optim. 12(5–6), 535–543 (1991)
Nakao, M.T., Yamamoto, N.: Numerical verification of solutions for nonlinear elliptic problems using an L ∞ residual method. J. Math. Anal. Appl. 217(1), 246–262 (1998)
Rump, S.M.: Verification methods: rigorous results using floating-point arithmetic. Acta Numer. 19, 287–449 (2010)
Toyonaga, K., Nakao, M.T., Watanabe, Y.: Verified numerical computations for multiple and nearly multiple eigenvalues of elliptic operators. J. Comput. Appl. Math. 147(1), 175–190 (2002)
Watanabe, Y., Nakao, M.T.: Numerical verifications of solutions for nonlinear elliptic equations. Japan J. Indust. Appl. Math. 10(1), 165–178 (1993)
Watanabe, Y., Yamamoto, N., Nakao, M.T.: Verified computations of solutions for nondifferentiable elliptic equations related to MHD equilibria. Nonlinear Anal. 28(3), 577–587 (1997)
Watanabe, Y., Yamamoto, N., Nakao, M.T.: An efficient approach to the numerical verification for solutions of elliptic differential equations with local uniqueness (in japanese). Trans. Japan Soc. Ind. Appl. Math. 15(4), 509–520 (2005)
Yamamoto, N.: A numerical verification method for solutions of boundary value problems with local uniqueness by Banach’s fixed-point theorem. SIAM J. Numer. Anal. 35(5), 2004–2013 (1998)
Yamamoto, N., Nakao, M.T.: Numerical verifications of solutions for elliptic equations in nonconvex polygonal domains. Numer. Math. 65(4), 503–521 (1993)
Yamamoto, N., Nakao, M.T.: Numerical verifications for solutions to elliptic equations using residual iterations with a higher order finite element. J. Comput. Appl. Math. 60(1–2), 271–279 (1995). Linear/Nonlinear Iterative Methods and Verification of Solution, Matsuyama, 1993
Zeidler, E.: Nonlinear Functional Analysis and Its Applications. I. Springer, New York (1986). Fixed-point theorems, Translated from the German by Peter R. Wadsack
Author information
Authors and Affiliations
Rights and permissions
Copyright information
© 2019 Springer Nature Singapore Pte Ltd.
About this chapter
Cite this chapter
Nakao, M.T., Plum, M., Watanabe, Y. (2019). Newton-Type Approaches in Finite Dimension. In: Numerical Verification Methods and Computer-Assisted Proofs for Partial Differential Equations. Springer Series in Computational Mathematics, vol 53. Springer, Singapore. https://doi.org/10.1007/978-981-13-7669-6_2
Download citation
DOI: https://doi.org/10.1007/978-981-13-7669-6_2
Published:
Publisher Name: Springer, Singapore
Print ISBN: 978-981-13-7668-9
Online ISBN: 978-981-13-7669-6
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)