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On the approximation of capillary surfaces in a gravitational field

Zur Approximation von Kapillarflächen in einem Gravitationsfeld

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Abstract

The shape of the surface of a liquid inside a cylindrical tube of general cross-section is approximated by the simplest finite element method. Some a priori estimates are obtained for the approximate solution. Then convergence including pointwise estimates is proved as for uniformly elliptic problems. The analysis also covers the case with a volume contraint. Some numerical results are given.

Zusammenfassung

Die Form der Flüssigkeitsoberfläche in einem zylindrischen Rohr mit allgemeinem Querschnitt wird mit Hilfe der einfachsten Methode der finiten Elemente approximiert. Einige a priori Abschätzungen für die Näherungslösung werden gezeigt. Dann wird Konvergenz einschließlich gleichmäßiger Abschätzungen wie bei gleichmäßig elliptischen Problemen bewiesen. Der Fall festen Flüssigkeitsvolumens ist hierbei eingeschlossen. Es werden einige numerische Ergebnisse angegeben.

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References

  1. Concus, P.: Static menisci in a vertical right circular cylinder. J. Fluid Mech.34, 481–495 (1968).

    Google Scholar 

  2. Concus, P., Finn, R.: On the behavior of a capillary surface in a wedge. Proc. Nat. Acad. Sci.63, 292–299 (1969).

    Google Scholar 

  3. Concus, P., Finn, R.: On capillary free surfaces in a gravitational field. Act. Math.132, 207–223 (1974).

    Google Scholar 

  4. Gerhardt, C.: Global regularity of the solutions to the capillarity problem. Ann. Scuola Norm. Sup. Pisa Cl. Sci.III, 157–175 (1976).

    Google Scholar 

  5. Il'in, V. P.: Some inequalities in function spaces and their application to the study of the convergence of variational processes. Amer. Math. Soc. Transl.81, 1–66 (1969).

    Google Scholar 

  6. Johnson, C., Thomee, V.: Error estimates for a finite element approximation of a minimal surface. Math. Comp.20, 343–349 (1975).

    Google Scholar 

  7. Johnson, W. E., Perko, L. M.: Interior and exterior boundary value problems from the theory of the capillary tube. Arch. Rat. Mech. Anal.29, 125–143 (1968).

    Google Scholar 

  8. Mittelmann, H. D.: Die Approximation der Lösungen gemischter Randwertprobleme quasilinearer elliptischer Differentialgleichungen. Computing13, 253–265 (1974).

    Google Scholar 

  9. Mittelmann, H. D.: On pointwise estimates for a finite element solution of nonlinear boundary value problems. SIAM J. Num. Anal. (to appoar).

  10. Mittelmann, H. D.: Numerische Behandlung nichtlinearer Randwertprobleme mit finiten Elementen. Computing18, 67–77 (1977).

    Google Scholar 

  11. Necas, J.: Les méthodes directes en théorie des équations elliptiques. Prag: Academia 1967.

    Google Scholar 

  12. Strang, G., Fix, G. J.: An analysis of the finite element method. Englewood Cliffs, N. J.: Prentice-Hall 1973.

    Google Scholar 

  13. Uralt'seva, N. N.: Nonlinear boundary value problems for equations of minimal-surface type. Proc. Steklov Inst. Math.116, 227–237 (1971).

    Google Scholar 

  14. Uralt'seva, N. N.: The solvability of the capillarity problem. Vestnik Leningrad Univ. No. 19 Mat. Met. Astronom. Vyp.4, 54–64 (1973). (Russian).

    Google Scholar 

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Authors and Affiliations

  1. Fachbereich Mathematik, Technische Hochschule Darmstadt, Kantplatz 1, D-6100, Darmstadt, Federal Republic of Germany

    H. D. Mittelmann

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  1. H. D. Mittelmann
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Mittelmann, H.D. On the approximation of capillary surfaces in a gravitational field. Computing 18, 141–148 (1977). https://doi.org/10.1007/BF02243623

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  • DOI: https://doi.org/10.1007/BF02243623

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