This paper is devoted to the numerical solution of nonlinear elliptic partial differential equations. Such problems describe various phenomena in science. An approach that exploits Hilbert space theory in the numerical study of elliptic PDEs is the idea of preconditioning operators. In this survey paper we briefly summarize the main lines of this theory with various applications.
@article{bwmeta1.element.doi-10_2478_s11533-011-0060-9,
author = {Janos Kar\'atson},
title = {Operator preconditioning with efficient applications for nonlinear elliptic problems},
journal = {Open Mathematics},
volume = {10},
year = {2012},
pages = {231-249},
zbl = {1247.65146},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.doi-10_2478_s11533-011-0060-9}
}
Janos Karátson. Operator preconditioning with efficient applications for nonlinear elliptic problems. Open Mathematics, Tome 10 (2012) pp. 231-249. http://gdmltest.u-ga.fr/item/bwmeta1.element.doi-10_2478_s11533-011-0060-9/
[1] Antal I., Karátson J., A mesh independent superlinear algorithm for some nonlinear nonsymmetric elliptic systems, Comput. Math. Appl., 2008, 55(10), 2185–2196 http://dx.doi.org/10.1016/j.camwa.2007.11.014 | Zbl 1142.65439
[2] Antal I., Karátson J., Mesh independent superlinear convergence of an inner-outer iterative method for semilinear elliptic interface problems, J. Comput. Appl. Math., 2009, 226(2), 190–196 http://dx.doi.org/10.1016/j.cam.2008.08.001 | Zbl 1163.65071
[3] Axelsson O., On global convergence of iterative methods, In: Iterative Solution of Nonlinear Systems of Equations, Oberwolfach, 1982, Lecture Notes in Math., 953, Springer, Berlin-New York, 1982, 1–19 http://dx.doi.org/10.1007/BFb0069371
[4] Axelsson O., Iterative Solution Methods, Cambridge University Press, Cambridge, 1994 | Zbl 0795.65014
[5] Axelsson O., Faragó I., Karátson J., Sobolev space preconditioning for Newton’s method using domain decomposition, Numer. Linear Algebra Appl., 2002, 9(6–7), 585–598 http://dx.doi.org/10.1002/nla.293 | Zbl 1071.65547
[6] Axelsson O., Karátson J., Conditioning analysis of separate displacement preconditioners for some nonlinear elasticity systems, Math. Comput. Simulation, 2004, 64(6), 649–668 http://dx.doi.org/10.1016/j.matcom.2003.11.017 | Zbl 1088.74013
[7] Axelsson O., Karátson J., Equivalent operator preconditioning for elliptic problems, Numer. Algorithms, 2009, 50(3), 297–380 http://dx.doi.org/10.1007/s11075-008-9233-4 | Zbl 1162.65024
[8] Deuflhard P., Weiser M., Global inexact Newton multilevel FEM for nonlinear elliptic problems, In: Multigrid Methods V, Stuttgart, 1996, Lecture Notes Comput. Sci. Eng., 3, Springer, Berlin, 1998, 71–89 http://dx.doi.org/10.1007/978-3-642-58734-4_4
[9] Faber V., Manteuffel T.A., Parter S.V., On the theory of equivalent operators and application to the numerical solution of uniformly elliptic partial differential equations, Adv. in Appl. Math., 1990, 11(2), 109–163 http://dx.doi.org/10.1016/0196-8858(90)90007-L | Zbl 0718.65043
[10] Faragó I., Karátson J., Numerical Solution of Nonlinear Elliptic Problems via Preconditioning Operators: Theory and Applications, Adv. Comput. Theory Pract., 11, Nova Science Publishers, Hauppauge, 2002 | Zbl 1030.65117
[11] Faragó I., Karátson J., Sobolev gradient type preconditioning for the Saint-Venant model of elasto-plastic torsion, Int. J. Numer. Anal. Model., 2008, 5(2), 206–221 | Zbl 1138.35089
[12] Kantorovich L.V., Akilov G.P., Functional Analysis, 2nd ed., Pergamon Press, Oxford-Elmsford, 1982 | Zbl 0484.46003
[13] Křížek M., Neittaanmäki P., Mathematical and Numerical Modelling in Electrical Engineering, Math. Model. Theory Appl., 1, Kluwer, Dordrecht, 1996 | Zbl 0859.65128
[14] Karátson J., Gradient method in Sobolev space for nonlocal boundary value problems, Electron. J. Differential Equations, 2000, #51 | Zbl 0995.35020
[15] Karátson J., Constructive Sobolev gradient preconditioning for semilinear elliptic systems, Electron. J. Differential Equations, 2004, #75 | Zbl 1109.65308
[16] Karátson J., Faragó I., Variable preconditioning via quasi-Newton methods for nonlinear problems in Hilbert space, SIAM J. Numer. Anal., 2003, 41(4), 1242–1262 http://dx.doi.org/10.1137/S0036142901384277 | Zbl 1130.65309
[17] Karátson J., Kurics T., Lirkov I., A parallel algorithm for systems of convection-diffusion equations, In: Numerical Methods and Applications, 2006, Lecture Notes in Comput. Sci., 4310, Springer, New York, 2007, 65–73 http://dx.doi.org/10.1007/978-3-540-70942-8_7
[18] Karátson J., Lóczi L., Sobolev gradient preconditioning for the electrostatic potential equation, Comput. Math. Appl., 2005, 50(7), 1093–1104 http://dx.doi.org/10.1016/j.camwa.2005.08.011 | Zbl 1098.65113
[19] Karátson J., Neuberger J.W., Newton’s method in the context of gradients, Electron. J. Differential Equations, 2007, #124 | Zbl 1135.65337
[20] Keller H.B., Elliptic boundary value problems suggested by nonlinear diffusion processes, Arch. Rational Mech. Anal., 1969, 35(5), 363–381 http://dx.doi.org/10.1007/BF00247683 | Zbl 0188.17102
[21] Kovács B., A comparison of some efficient numerical methods for a nonlinear elliptic problem, Cent. Eur. J. Math., 2012, 10(1)
[22] Martinsson P.-G., A fast direct solver for a class of elliptic partial differential equations, J. Sci. Comput., 2009, 38(3), 316–330 http://dx.doi.org/10.1007/s10915-008-9240-6 | Zbl 1203.65066
[23] Meurant G., Computer Solution of Large Linear Systems, Stud. Math. Appl., 28, North-Holland, Amsterdam, 1999
[24] Neuberger J.W., Prospects for a central theory of partial differential equations, Math. Intelligencer, 2005, 27(3), 47–55 http://dx.doi.org/10.1007/BF02985839 | Zbl 1090.35011
[25] Neuberger J.W., Sobolev Gradients and Differential Equations, 2nd ed., Lecture Notes in Math., 1670, Springer, Berlin, 2010 http://dx.doi.org/10.1007/978-3-642-04041-2 | Zbl 1203.35004
[26] Neuberger J.W., Renka R.J., Sobolev gradients and the Ginzburg-Landau functional, SIAM J. Sci. Comput., 1998, 20(2), 582–590 http://dx.doi.org/10.1137/S1064827596302722 | Zbl 0920.35059
[27] Neuberger J.W., Renka R.J., Sobolev gradients: introduction, applications, problems, In: Variational Methods: Open Problems, Recent Progress, and Numerical Algorithms, Contemp. Math., 257, American Mathematical Society, Providence, 2004, 85–99 | Zbl 1064.35006
[28] Richardson W.B., Jr., Sobolev gradient preconditioning for image-processing PDEs, Comm. Numer. Methods Eng., 2008, 24(6), 493–504 http://dx.doi.org/10.1002/cnm.951 | Zbl 1168.68596
[29] Szabó B., Babuška I., Finite Element Analysis, Wiley-Intersci. Publ., John Wiley & Sons, New York, 1991 | Zbl 0792.73003
[30] Zlatev Z., Computer Treatment of Large Air Pollution Models, Environmental Science and Technology Library, 2, Kluwer, Dordrecht-Boston-London, 1995 http://dx.doi.org/10.1007/978-94-011-0311-4