Abstract
For elliptic boundary value problems of the form −ΔU+F(x, U, U x )=0 on Ω,B[U]=0 on ϖΩ, with a nonlinearityF growing at most quadratically with respect to the gradientU x and with a mixed-type linear boundary opeatorB, a numerical method is presented which can be used to prove the existence of a solution within a “close”H 1,4(Ω)-neighborhood of some approximate solution ω∈H 2(Ω) satisfying the boundary condition, provided that the defect-norm ∥−Δω +F(·, ω, ω x )∥2 is sufficiently small and, moreover, the linearization of the given problem at ω leads to an invertible operatorL. The main tools are explicit Sobolev imbeddings and eigenvalue bounds forL or forL*L. All kinds of monotonicity or inverse-positivity assumptions are avoided.
Zusammenfassung
Gegeben sei ein elliptisches Randwertproblem der Form −ΔU+F(x, U, U x )=0 auf Ω,B[U]=0 auf ϖΩ, mit einer NichtlinearitätF, die einer quadratischen Wachstumsbedingung bezüglich des GradientenU x genügt, und mit einem linearen RandoperatorB von gemischtem Typ. Es wird eine numerische Methode vorgestellt, mit deren Hilfe sich die Existenz einer Lösung innerhalb einer “kleinen”H 1,4(Ω)-Umgebung einer Näherungslösung ω∈H 2(Ω), die die Randbedingung erfüllt, nachweisen läßt, sofern die Defektnorm ∥−Δω +F(·, ω, ω x )∥2 hinreichend klein ist und ferner die Linearisierung des gegebenen Problems in ω auf einen invertierbaren OperatorL führt. Die wesentlichen Hilfsmittel sind explizite Sobolevsche Einbettungen und Eigenwertschranken fürL oderL*L. Jegliche Monotonie- und Inverspositivitätsbedingungen werden vermieden.
Similar content being viewed by others
Explore related subjects
Discover the latest articles, news and stories from top researchers in related subjects.References
Adams, R. A.: Sobolev spaces. New York: Academic Press 1975.
Bauer, H.: Wahrscheinlichkeitstheorie und Grundzüge der Maßtheorie, 3rd edn.. Berlin:de Gruyter 1978.
Bazley, N. W., Fox, D. W.: Comparison operators for lower bounds to eigenvalues.J. Reine Angew. Math.223, 142–149 (1966).
Collatz, L.: The numerical treatment of differential equations. Berlin, Göttingen, Heidelberg: Springer 1960.
Ehlich, H., Zeller, K.: Schwankung von Polynomen zwischen Gitterpunkten. Math. Z.86, 41–44 (1964).
Friedman, A.: Partial differential equations. New York: Holt, Rinehart and Winston 1969.
Gilbarg, D., Trudinger, N. S.: Elliptic partial differential equations of second order, 2nd edn. Berlin, Heidelberg: Springer 1983.
Göhlen, M., Plum, M., Schröder, J.: A programmed algorithm for existence proofs for two-point boundary value problems. Computing44, 91–132 (1990).
Goerisch, F., Albrecht, J.: Eine einheitliche Herleitung von Einschließungsätzen für Eigenwerte. In: Albrecht, J., Collatz, L., Velte, W. (eds.) Numerical treatment of eigenvalue problems, vol. 3, 58–88, ISNM 69, Basel: Birkhäuser 1984.
Grisvard, P.: Elliptic problems in nonsmooth domains. Boston: Pitman Publ. 1985.
IBM High-Accuracy Arithmetic Subroutine Library (ACRITH). Program description and user's guide, SC 33-6164-02, 3rd edition (1986).
Kulisch, U.: FORTRAN-SC, language reference and user's guide. University of Karlsruhe and IBM Development Laboratory Böblingen, 1987.
Ladyzhenskaya, O. A., Ural'tseva, N. N.: Linear and quasilinear elliptic equations. New York: Academic Press 1968.
Nečas, J.: Les méthodes directes en théorie des équations elliptiques. Paris: Masson et cie. 1967.
Plum, M.: Verified existence and inclusion results for two-point boundary value problems. IMACS Annals on Computing and Applied Mathematics,7, 341–355, Baltzer 1990.
Plum, M.: Computer-assisted existence proofs for two-point boundary value problems. Computing46, 19–34 (1991).
Plum, M.: ExplicitH 2-estimates and pointwise bounds for solutions of second-order elliptic boundary value problems. J. Math. Anal. Appl.165, 36–61 (1992).
Plum, M.: Existence proofs in combination with error bounds for approximate solutions of weakly nonlinear second-order elliptic boundary value problems. ZAMM71, T660-T662 (1991).
Plum, M.: Bounds for eigenvalues of second-order elliptic differential operators. J. Appl. Math. Physics (ZAMP)42, 848–863 (1991).
Schröder, J.: Operator inequalities. New York: Academic Press 1980.
Schröder, J.: Existence proofs for boundary value problems by numerical algorithms Report Univ. Cologne 1986.
Schröder, J.: A method for producing verified results for two-point boundary value problems. Computing [Suppl.]6, 9–22 (1988).
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Plum, M. Numerical existence proofs and explicit bounds for solutions of nonlinear elliptic boundary value problems. Computing 49, 25–44 (1992). https://doi.org/10.1007/BF02238648
Received:
Issue Date:
DOI: https://doi.org/10.1007/BF02238648