Abstract
A generalized monotonicity theorem in the sense of Collatz is proved for almost linear elliptic boundary value problems of second order. This theorem is valid for continuous spline functions as used by the finite element method — in contrast to the classical theorems. The background is a generalized maximum principle with an interface condition for the normal derivatives which has been proven by Natterer/Werner in the case of the Laplace operator. Some numerical examples providing pointwise bounds for the solution of Dirichlet problems by cubic splines, the so called T 10 elements, are given.
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
Preview
Unable to display preview. Download preview PDF.
Literatur
Collatz, L.: Aufgaben monotoner Art, Arch. Math. 3 (1952), 366–376.
Collatz, L.: Vortrag auf dem Symposium über “Numerische Methoden bei partiellen Differentialgleichungen” in Keszthely, Ungarn, September 1973, erscheint demnächst.
Littmann, W.: A strong maximum principle for weakly L-subharmonic functions, J. of Math. and Mech. 8 (1959), 761–770.
Littmann, W.: Generalized subharmonic functions: monotonic approximations and an improved maximum principle, Annali delle Scuola Norm. Sup. di Pisa., (3) 17 (1963), 207–222.
Natterer, F. und B. Werner: Verallgemeinerung des Maximumprinzips für den Laplace-Operator. Num. Math. 22 (1974), 149–156.
Protter, M.H. und H.F. Weinberger: Maximum principles in differential equations, Prentice-Hall, (1967).
Redheffer, R.: Bemerkungen über Monotonie und Fehlerabschätzungen bei nichtlinearen partiellen Differentialgleichungen, Arch. for Rat. Mech. and Anal. 10 (1962), 427–457.
Schräder, J.: Upper and lower bounds for the solutions of two-point boundary value problems. (1974); Erscheint demnächst.
Strang, G. und G.J. Fix: An analysis of the finite element method, Prentice-Hall (1973).
Werner, B.: Verallgemeinerte Monotonie mit Anwendungen auf Spline-Funktionen. Habilitationsschrift an der Universität Hamburg, (1974).
Westphal, H.: Zur Abschätzung der Lösungen nichtlinearer parabolischer Differentialgleichungen, Math. Zeitschr. 51, (1949), 690–695.
Wetterling, W.: Lösungsschranken bei elliptischen Differentialgleichungen ISNM 9 (1968), 393–401, BirkhäuserVerlag.
Author information
Authors and Affiliations
Rights and permissions
Copyright information
© 1975 Springer Basel AG
About this chapter
Cite this chapter
Werner, B. (1975). Monotonie und Finite Elemente bei Elliptischen Differentialgleichungen. In: Numerische Behandlung von Differentialgleichungen. International Series of Numerical Mathematics / Internationale Schriftenreihe zur Numerischen Mathematik / Série Internationale D’Analyse Numérique, vol 27. Birkhäuser, Basel. https://doi.org/10.1007/978-3-0348-5532-7_20
Download citation
DOI: https://doi.org/10.1007/978-3-0348-5532-7_20
Publisher Name: Birkhäuser, Basel
Print ISBN: 978-3-0348-5533-4
Online ISBN: 978-3-0348-5532-7
eBook Packages: Springer Book Archive