Error estimates of an efficient linearization scheme for a nonlinear elliptic problem with a nonlocal boundary condition
ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique, Volume 35 (2001) no. 4, p. 691-711

We consider a nonlinear second order elliptic boundary value problem (BVP) in a bounded domain $\Omega \subset {ℝ}^{dim}$ with a nonlocal boundary condition. A Dirichlet BC containing an unknown additive constant, accompanied with a nonlocal (integral) Neumann side condition is prescribed at some boundary part ${\Gamma }_{n}$. The rest of the boundary is equipped with Dirichlet or nonlinear Robin type BC. The solution is found via linearization. We design a robust and efficient approximation scheme. Error estimates for the linearization algorithm are derived in ${L}_{2}\left(\Omega \right),{H}^{1}\left(\Omega \right)$ and ${L}_{\infty }\left(\Omega \right)$ spaces.

Classification:  65N15,  35J60
Keywords: nonlinear elliptic BVP, error estimates, nonstandard boundary condition, linearization
@article{M2AN_2001__35_4_691_0,
author = {Slodi\v cka, Marian},
title = {Error estimates of an efficient linearization scheme for a nonlinear elliptic problem with a nonlocal boundary condition},
journal = {ESAIM: Mathematical Modelling and Numerical Analysis - Mod\'elisation Math\'ematique et Analyse Num\'erique},
publisher = {EDP-Sciences},
volume = {35},
number = {4},
year = {2001},
pages = {691-711},
zbl = {0997.65124},
mrnumber = {1862875},
language = {en},
url = {http://www.numdam.org/item/M2AN_2001__35_4_691_0}
}

Slodička, Marian. Error estimates of an efficient linearization scheme for a nonlinear elliptic problem with a nonlocal boundary condition. ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique, Volume 35 (2001) no. 4, pp. 691-711. http://www.numdam.org/item/M2AN_2001__35_4_691_0/

[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

[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

[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

[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

[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

[6] L.C. Evans, Partial differential equations, Graduate Studies in Mathematics 19, American Mathematical Society (1998). | MR 1625845 | Zbl 0902.35002

[7] A. Friedman, Variational principles and free-boundary problems. Wiley, New York (1982). | MR 679313 | 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

[9] D. Gilbarg and N.S. Trudinger, Elliptic Partial Differential Equations of Second Order. Springer, Berlin, Heidelberg (1983). | MR 737190 | 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

[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

[12] C.V. Pao, Nonlinear parabolic and elliptic equations. Plenum Press, New York (1992). | MR 1212084 | 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

[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

[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

[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

[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

[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.