Operator preconditioning with efficient applications for nonlinear elliptic problems

 Access to full text

 Full (PDF)

1. [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 Zbl1142.65439
2. [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 Zbl1163.65071
3. [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. [4] Axelsson O., Iterative Solution Methods, Cambridge University Press, Cambridge, 1994 Zbl0795.65014
5. [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 Zbl1071.65547
6. [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 Zbl1088.74013
7. [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 Zbl1162.65024
8. [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. [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 Zbl0718.65043
10. [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 Zbl1030.65117
11. [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 Zbl1138.35089
12. [12] Kantorovich L.V., Akilov G.P., Functional Analysis, 2nd ed., Pergamon Press, Oxford-Elmsford, 1982 Zbl0484.46003
13. [13] Křížek M., Neittaanmäki P., Mathematical and Numerical Modelling in Electrical Engineering, Math. Model. Theory Appl., 1, Kluwer, Dordrecht, 1996 Zbl0859.65128
14. [14] Karátson J., Gradient method in Sobolev space for nonlocal boundary value problems, Electron. J. Differential Equations, 2000, #51 Zbl0995.35020
15. [15] Karátson J., Constructive Sobolev gradient preconditioning for semilinear elliptic systems, Electron. J. Differential Equations, 2004, #75 Zbl1109.65308
16. [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 Zbl1130.65309
17. [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. [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 Zbl1098.65113
19. [19] Karátson J., Neuberger J.W., Newton’s method in the context of gradients, Electron. J. Differential Equations, 2007, #124 Zbl1135.65337
20. [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 Zbl0188.17102
21. [21] Kovács B., A comparison of some efficient numerical methods for a nonlinear elliptic problem, Cent. Eur. J. Math., 2012, 10(1) 
22. [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 Zbl1203.65066
23. [23] Meurant G., Computer Solution of Large Linear Systems, Stud. Math. Appl., 28, North-Holland, Amsterdam, 1999 
24. [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 Zbl1090.35011
25. [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 Zbl1203.35004
26. [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 Zbl0920.35059
27. [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 Zbl1064.35006
28. [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 Zbl1168.68596
29. [29] Szabó B., Babuška I., Finite Element Analysis, Wiley-Intersci. Publ., John Wiley & Sons, New York, 1991 Zbl0792.73003
30. [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