Hostname: page-component-6d856f89d9-4thr5 Total loading time: 0 Render date: 2024-07-16T07:45:07.847Z Has data issue: false hasContentIssue false
  • English
  • Français

A semi-smooth Newton method for solving elliptic equations with gradient constraints

Published online by Cambridge University Press:  05 December 2008

Roland Griesse  and
Karl Kunisch
Roland Griesse
Affiliation:
Chemnitz University of Technology, Faculty of Mathematics, 09107 Chemnitz, Germany. roland.griesse@mathematik.tu-chemnitz.de; http://www.tu-chemnitz.de/ griesse
Karl Kunisch
Affiliation:
Institute for Mathematics and Scientific Computing, Heinrichstraße 36, 8010 Graz, Austria. karl.kunisch@uni-graz.at; http://www.kfunigraz.ac.at/imawww/kunisch
Get access

Abstract

Semi-smooth Newton methods for elliptic equations with gradient constraints are investigated. The one- and multi-dimensional cases are treated separately. Numerical examples illustrate the approach and as well as structural features of the solution.

Keywords

Gradient constraintsactive set strategyregularizationsemi-smooth Newton methodprimal-dual active set method.
Type
Research Article
Information
ESAIM: Mathematical Modelling and Numerical Analysis , Volume 43 , Issue 2 , March 2009 , pp. 209 - 238
Copyright
© EDP Sciences, SMAI, 2009

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Bony, J.-M., Principe du maximum dans les espaces de Sobolev. C. R. Acad. Sci. Paris Sér. A-B 265 (1967) 333–336.
Brooks, A. and Hughes, T., Streamline upwind/Petrov-Galerkin formulations for convection dominated flows with particular emphasis on the incompressible Navier-Stokes equations. Comput. Methods Appl. Mech. Eng. 32 (1982) 99–259. CrossRef
Chen, X., Superlinear convergence and smoothing quasi-Newton methods for nonsmooth equations. J. Comput. Appl. Math. 80 (1997) 105–126.
M. Delfour and J.-P. Zolésio, Shapes and Geometries. Analysis, Differential Calculus, and Optimization. Philadelphia (2001).
Evans, L.C., A second order elliptic equation with gradient constraint. Comm. Partial Differ. Equ. 4 (1979) 555–572. CrossRef
D. Gilbarg and N.S. Trudinger, Elliptic Differential Equations of Second Order. Springer, New York (1977).
M. Hintermüller and K. Kunisch, Stationary optimal control problems with pointwise state constraints. SIAM J. Optim. (to appear).
Hintermüller, M., Ito, K. and Kunisch, K., The primal-dual active set strategy as a semismooth Newton method. SIAM J. Optim. 13 (2002) 865–888.
Ishii, H. and Koike, S., Boundary regularity and uniqueness for an elliptic equation with gradient constraint. Comm. Partial Differ. Equ. 8 (1983) 317–346.
Ito, K. and Kunisch, K., The primal-dual active set method for nonlinear optimal control problems with bilateral constraints. SIAM J. Contr. Opt. 43 (2004) 357–376.
C. Johnson, Numerical Solution of Partial Differential Equations by the Finite Element Method. Cambridge University Press, Cambridge (1987).
K. Kunisch and J. Sass, Trading regions under proportional transaction costs, in Operations Research Proceedings, U.M. Stocker and K.-H. Waldmann Eds., Springer, New York (2007) 563–568.
O.A. Ladyzhenskaya and N.N. Ural'tseva, Linear and Quasilinear Elliptic Equations. Academic Press, New York (1968).
Shreve, S. and Soner, H.M., Optimal investment and consumption with transaction costs. Ann. Appl. Probab. 4 (1994) 609–692. CrossRef
K. Stromberg, Introduction to Classical Real Analysis. Wadsworth International, Belmont, California (1981).
G. Troianiello, Elliptic Differential Equations and Obstacle Problems. Plenum Press, New York (1987).
Wiegner, M., The C 1,1-character of solutions of second order elliptic equations with gradient constraint. Comm. Partial Differ. Equ. 6 (1981) 361–371. CrossRef