| |
Table of contents for this issue | Previous article | Next article
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, 35 no. 4 (2001), p. 691-711
Full text djvu | pdf | Reviews MR 1862875 | Zbl 0997.65124
Class. Math.: 65N15, 35J60
Keywords: nonlinear elliptic BVP, error estimates, nonstandard boundary condition, linearization
stable URL: http://www.numdam.org/item?id=M2AN_2001__35_4_691_0
We consider a nonlinear second order elliptic boundary value problem (BVP) in a bounded domain $\Omega \subset {\mathbb{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 $L_{2} (\Omega ),H^{1}(\Omega )$ and $L_{\infty } (\Omega )$ spaces.
[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. |
|
Copyright Cellule MathDoc 2005 | Credit | Site Map
|