Error estimates of an efficient linearization scheme for a nonlinear elliptic problem with a nonlocal boundary condition

Slodička, Marian

ESAIM: Modélisation mathématique et analyse numérique, Tome 35 (2001) no. 4, pp. 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
Mots clés : nonlinear elliptic BVP, error estimates, nonstandard boundary condition, linearization

[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

[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

[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

[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 | Zbl

[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

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

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

[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 | Zbl

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

[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

[11] J. Kačur, Solution to strongly nonlinear parabolic problems by a linear approximation scheme. IMA J. Numer. Anal. 19 (1999) 119-145. | Zbl

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

[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

[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

[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 | Zbl

[16] M. Slodička and H. De Schepper, A nonlinear boundary value problem containing nonstandard boundary conditions. Appl. Math. Comput. (to appear). | MR | Zbl

[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

[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.