Equadiff 6 pp 327-332 | Cite as

Free boundary problems for stokes' flows and finite element methods

  • J. A. Nitsche
Lectures Presented In Sections Section C Numerical Methods
Part of the Lecture Notes in Mathematics book series (LNM, volume 1192)


In two dimensions a Stokes' flow is considered symmetric to the abscissa η=0 and periodic with respect to ξ. On the free boundary |η|=S(ξ) the conditions are: (i) the free boundary is a streamline, (ii) the tangential force vanishes, (iii) the normal force is proportional to the mean curvature of the boundary. By straightening the boundary, i. e. by introducing the variables x=ξ, y=η/S(ξ), the problem is reduced to one in a fixed domain. The underlying differential equations are now highly nonlinear: They consist in an elliptic system coupled with an ordinary differential equation for S. The analytic properties of the solution as well as the convergence of the proposed finite element approximation are discussed.


Free Boundary Tangential Force Elliptic System Free Boundary Problem Approximation Space 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. Bemelmans, J. (1981a) Gleichgewichtsfiguren zäher Flüssigkeiten mit Oberflächenspannung Analysis 1, 241–282 (1981)MathSciNetCrossRefzbMATHGoogle Scholar
  2. Bemelmans, J. (1981b) Liquid Drops in a viscous Fluid under the Influence of Gravity and Surface Tension Manuscripta math. 36, 105–123 (1981)MathSciNetCrossRefzbMATHGoogle Scholar
  3. Bemelmans, J. and A. Friedman (1984) Analiticity for the Navier-Stokes Equations Governed by Surface Tension on the Free Boundary J. of Diff. Equat. 55, 135–150 (1984)MathSciNetCrossRefzbMATHGoogle Scholar
  4. Nitsche, J. A. Schauder Estimates for Finite Element Approximations on second Order Elliptic Boundary Value Problems Proceedings of the Special Year in Numerical Analysis, Lecture Notes #20, Univ. of Maryland, Babuska, I., T.,-P. Liu, and J. Osborn eds., 290–343 (1981)Google Scholar
  5. Schulz, F. (1982) Über elliptische Monge-Amperesche Differentialgleichungen mit einer Bemerkung zum Weylschen Einbettungsproblem Nachr. Akad. Wiss. Göttingen, II Math.-Phys. Klasse 1981, 93–108 (1982)zbMATHGoogle Scholar

Copyright information

© Equadiff 6 and Springer-Verlag 1986

Authors and Affiliations

  • J. A. Nitsche
    • 1
  1. 1.Institut für angewandte MathematikAlbert-Ludwigs-UniversitätFreiburg im BreisgauGermany

Personalised recommendations