A semi-smooth Newton method for solving elliptic equations with gradient constraints
ESAIM: Modélisation mathématique et analyse numérique, Tome 43 (2009) no. 2, pp. 209-238.

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.

DOI : 10.1051/m2an:2008049
Classification : 35J70, 49M15, 65K05, 90C33
Mots-clés : gradient constraints, active set strategy, regularization, semi-smooth Newton method, primal-dual active set method 
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     author = {Griesse, Roland and Kunisch, Karl},
     title = {A semi-smooth {Newton} method for solving elliptic equations with gradient constraints},
     journal = {ESAIM: Mod\'elisation math\'ematique et analyse num\'erique},
     pages = {209--238},
     publisher = {EDP-Sciences},
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     doi = {10.1051/m2an:2008049},
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     zbl = {1161.65338},
     language = {en},
     url = {http://www.numdam.org/articles/10.1051/m2an:2008049/}
}
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Griesse, Roland; Kunisch, Karl. A semi-smooth Newton method for solving elliptic equations with gradient constraints. ESAIM: Modélisation mathématique et analyse numérique, Tome 43 (2009) no. 2, pp. 209-238. doi : 10.1051/m2an:2008049. http://www.numdam.org/articles/10.1051/m2an:2008049/

[1] J.-M. Bony, Principe du maximum dans les espaces de Sobolev. C. R. Acad. Sci. Paris Sér. A-B 265 (1967) 333-336. | MR | Zbl

[2] A. Brooks and T. Hughes, 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. | MR | Zbl

[3] X. Chen, Superlinear convergence and smoothing quasi-Newton methods for nonsmooth equations. J. Comput. Appl. Math. 80 (1997) 105-126. | MR | Zbl

[4] M. Delfour and J.-P. Zolésio, Shapes and Geometries. Analysis, Differential Calculus, and Optimization. Philadelphia (2001). | MR | Zbl

[5] L.C. Evans, A second order elliptic equation with gradient constraint. Comm. Partial Differ. Equ. 4 (1979) 555-572. | MR | Zbl

[6] D. Gilbarg and N.S. Trudinger, Elliptic Differential Equations of Second Order. Springer, New York (1977). | MR | Zbl

[7] M. Hintermüller and K. Kunisch, Stationary optimal control problems with pointwise state constraints. SIAM J. Optim. (to appear). | MR

[8] M. Hintermüller, K. Ito and K. Kunisch, The primal-dual active set strategy as a semismooth Newton method. SIAM J. Optim. 13 (2002) 865-888. | MR | Zbl

[9] H. Ishii and S. Koike, Boundary regularity and uniqueness for an elliptic equation with gradient constraint. Comm. Partial Differ. Equ. 8 (1983) 317-346. | MR | Zbl

[10] K. Ito and K. Kunisch, The primal-dual active set method for nonlinear optimal control problems with bilateral constraints. SIAM J. Contr. Opt. 43 (2004) 357-376. | MR | Zbl

[11] C. Johnson, Numerical Solution of Partial Differential Equations by the Finite Element Method. Cambridge University Press, Cambridge (1987). | MR | Zbl

[12] 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. | Zbl

[13] O.A. Ladyzhenskaya and N.N. Ural'Tseva, Linear and Quasilinear Elliptic Equations. Academic Press, New York (1968). | Zbl

[14] S. Shreve and H.M. Soner, Optimal investment and consumption with transaction costs. Ann. Appl. Probab. 4 (1994) 609-692. | MR | Zbl

[15] K. Stromberg, Introduction to Classical Real Analysis. Wadsworth International, Belmont, California (1981). | MR | Zbl

[16] G. Troianiello, Elliptic Differential Equations and Obstacle Problems. Plenum Press, New York (1987). | MR | Zbl

[17] M. Wiegner, The C 1,1 -character of solutions of second order elliptic equations with gradient constraint. Comm. Partial Differ. Equ. 6 (1981) 361-371. | MR | Zbl

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