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Error estimates of an efficient linearization scheme for a nonlinear elliptic problem with a nonlocal boundary condition. (English) Zbl 0997.65124
The author develops a robust and efficient linearization algorithm for nonlinear elliptic second-order boundary value problems with mixed boundary conditions, including some nonstandard nonlocal conditions. The nonlinearity can appear as a source term or at Robin type boundary condition, are monotonically increasing and globally Lipschitz continuous and may degenerate at certain points. The author proves error estimates in several norms for the iterative scheme which converges for arbitrary starting points. This is illustrated for a numerical example on the unit square.

MSC:
65N15 Error bounds for boundary value problems involving PDEs
65N22 Numerical solution of discretized equations for boundary value problems involving PDEs
65N12 Stability and convergence of numerical methods for boundary value problems involving PDEs
35J65 Nonlinear boundary value problems for linear elliptic equations
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References:
[1] D. Andreucci and R. Gianni , Global existence and blow up in a parabolic problem with nonlocal dynamical boundary conditions . Adv. Differ. Equ. 1 ( 1996 ) 729 - 752 . Zbl 0852.35076 · Zbl 0852.35076
[2] D.N. Arnold and F. Brezzi , Mixed and nonconforming finite element methods: implementation, postprocessing and error estimates . RAIRO Modél. Math. Anal. Numér. 19 ( 1985 ) 7 - 32 . Numdam | Zbl 0567.65078 · Zbl 0567.65078
[3] J.H. Bramble and P. Lee , On variational formulations for the Stokes equations with nonstandard boundary conditions . RAIRO Modél. Math. Anal. Numér. 28 ( 1994 ) 903 - 919 . Numdam | Zbl 0819.76063 · Zbl 0819.76063
[4] H. Brézis , Opérateurs maximaux monotones et semi-groupes de contractions dans les espaces de Hilbert . North-Holland Math. Stud. 5, Notas de matemática 50, North-Holland Publishing Comp., Amsterdam, London; American Elsevier Publishing Comp. Inc., New York ( 1973 ). MR 348562 | Zbl 0252.47055 · Zbl 0252.47055
[5] H. De Schepper and M. Slodička , Recovery of the boundary data for a linear 2nd order elliptic problem with a nonlocal boundary condition . ANZIAM J. 42E ( 2000 ) C488-C505. ISSN 1442 - 4436 (formerly known as J. Austral. Math. Soc., Ser. B). Zbl 0977.65095 · Zbl 0977.65095
[6] L.C. Evans , Partial differential equations , Graduate Studies in Mathematics 19, American Mathematical Society ( 1998 ). MR 1625845 | Zbl 0902.35002 · Zbl 0902.35002
[7] A. Friedman , Variational principles and free-boundary problems . Wiley, New York ( 1982 ). MR 679313 | Zbl 0564.49002 · Zbl 0564.49002
[8] H. Gerke , U. Hornung , Y. Kelanemer , M. Slodička and S. Schumacher , Optimal Control of Soil Venting: Mathematical Modeling and Applications , ISNM 127, Birkhäuser, Basel ( 1999 ). MR 1686932 | Zbl 0919.73001 · Zbl 0919.73001
[9] D. Gilbarg and N.S. Trudinger , Elliptic Partial Differential Equations of Second Order . Springer, Berlin, Heidelberg ( 1983 ). MR 737190 | Zbl 0562.35001 · Zbl 0562.35001
[10] W. Jäger and J. Kačur , Solution of doubly nonlinear and degenerate parabolic problems by relaxation schemes . RAIRO Modél. Math. Anal. Numér. 29 ( 1995 ) 605 - 627 . Numdam | Zbl 0837.65103 · Zbl 0837.65103
[11] J. Kačur , Solution to strongly nonlinear parabolic problems by a linear approximation scheme . IMA J. Numer. Anal. 19 ( 1999 ) 119 - 145 . Zbl 0946.65145 · Zbl 0946.65145
[12] C.V. Pao , Nonlinear parabolic and elliptic equations . Plenum Press, New York ( 1992 ). MR 1212084 | Zbl 0777.35001 · Zbl 0777.35001
[13] R. Rannacher and S. Turek , Artificial boundaries and flux and pressure conditions for the incompressible Navier-Stokes equations . Internat. J. Numer. Methods Fluids 22 ( 1996 ) 325 - 352 . Zbl 0863.76016 · Zbl 0863.76016
[14] M. Slodička , A monotone linear approximation of a nonlinear elliptic problem with a non-standard boundary condition , in Algoritmy 2000, A. Handlovičová, M. Komorníková, K. Mikula and D. Ševčovič, Eds., Bratislava ( 2000 ) 47 - 57 . Zbl 1019.35032 · Zbl 1019.35032
[15] M. Slodička and H. De Schepper , On an inverse problem of pressure recovery arising from soil venting facilities . Appl. Math. Comput. (to appear). MR 1905411 | Zbl 1033.35145 · Zbl 1033.35145
[16] M. Slodička and H. De Schepper , A nonlinear boundary value problem containing nonstandard boundary conditions . Appl. Math. Comput. (to appear). MR 1920503 | Zbl 1135.35341 · Zbl 1135.35341
[17] M. Slodička and R. Van Keer , A nonlinear elliptic equation with a nonlocal boundary condition solved by linearization . Internat. J. Appl. Math. 6 ( 2001 ) 1 - 22 . Zbl 1030.35082 · Zbl 1030.35082
[18] R. Van Keer , L. Dupré and J. Melkebeek , Computational methods for the evaluation of the electromagnetic losses in electrical machinery . Arch. Comput. Methods Engrg. 5 ( 1999 ) 385 - 443 .
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