Hostname: page-component-7479d7b7d-wxhwt Total loading time: 0 Render date: 2024-07-12T01:30:58.989Z Has data issue: false hasContentIssue false
  • English
  • Français

A robust linearisation scheme for a nonlinear elliptic boundary value problem: Error estimates

Published online by Cambridge University Press:  17 February 2009

Marian Slodička
Marian Slodička
Affiliation:
Department of Mathematical Analysis, Ghent University, Galglaan 2, B-9000 Gent, Belgium; e-mail: marian.slodicka@ugent.be.
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

We consider a nonlinear second-order elliptic boundary value problem in a bounded domain Ω ⊂ RN with mixed boundary conditions. The solution is found via linearisation. We design a robust and efficient approximation scheme. Error estimates for the linearisation algorithm are derived in L2(Ω), H1(Ω) and L∞(Ω) spaces under the minimal regularity assumptions of the exact solution.

Type
Research Article
Information
The ANZIAM Journal , Volume 46 , Issue 4 , April 2005 , pp. 449 - 470
Copyright
Copyright © Australian Mathematical Society 2005

References

[1]Amann, H., “Supersolution, monotone iteration and stability”, J. Diff Eq. 21 (1976) 367–377.Google Scholar
[2]Arnold, D. N. and Brezzi, F., “Mixed and nonconforming finite element methods: implementation, postprocessing and error estimates”, RAIRO Modél. Math. Anal. Numér. 19 (1) (1985) 7–32.CrossRefGoogle Scholar
[3]Barrett, J. W. and Knabner, P., “Finite element approximation of the transport of reactive solutes in porous media. Part II: Error estimates for equilibrium adsorption processes”, SIAM J. Numer. Anal. 34 (1997) 455–479.CrossRefGoogle Scholar
[4]Deng, Y., Chen, G., Ni, W.-M. and Zhou, J., “Boundary element monotone iteration scheme for semilinear elliptic partial differential equations”, Math. Comput. 65 (1996) 943–982.Google Scholar
[5]Evans, L. C., Partial differential equations, Graduate Studies in Mathematics 19 (American Mathematical Society, Providence, RI, 1998).Google Scholar
[6]Jäger, W. and Kačur, J., “Solution of porous medium type systems by linear approximation schemes”, Numer Math. 60 (1991) 407–42.Google Scholar
[7]Jäger, W. and Kačur, J., “Solution of doubly nonlinear and degenerate parabolic problems by relaxation schemes”, RAIRO Modél. Math. Anal. Numér. 29 (5) (1995) 605–627.CrossRefGoogle Scholar
[8]Kačur, J., “Solution of some free boundary problems by relaxation schemes”, SIAM J. Numer Anal. 36 (1999) 290–316.CrossRefGoogle Scholar
[9]Kačur, J., “Solution to strongly nonlinear parabolic problems by a linear approximation scheme”, IMA J. Numer Anal. 19 (1) (1999) 119–145.CrossRefGoogle Scholar
[10]Křřžek, M. and Neittaanmäki, P., Mathematical and numerical modelling in electrical engineering. Theory and applications, Volume I of Mathematical modelling: Theory and applications, (Kluwer, Dordrecht, 1996).Google Scholar
[11]Maitre, E., “Numerical analysis of nonlinear elliptic-parabolic equations”, RAIRO Modél. Math. Anal. Numér. 36 (2002) 143–153.CrossRefGoogle Scholar
[12]Nečas, J., Introduction to the theory of nonlinear elliptic equations (John Wiley & Sons, Chichester, 1986).Google Scholar
[13]Pao, C. V., Nonlinear parabolic and elliptic equations (Plenum Press, New York, 1992).Google Scholar
[14]Pop, l. S. and Yong, W.-A., “On the existence and uniqueness of a solution for an elliptic problem”, Babes-Bolyayi 45 (2000) 97–107.Google Scholar
[15]Slodička, M., “Error estimates of an efficient linearization scheme for a nonlinear elliptic problem with a nonlocal boundary condition”, RAIRO Modél. Math. Anal. Numér. 35 (2001) 691–711.Google Scholar
[16]Slodička, M., “Mixed finite element method for nonlinear second-order elliptic problems: Relaxation scheme”, in Algoritmy 2002 (eds. Handlovićová, A., Krivá, Z., Mikula, K. and Śevčovič, D.), (Slovak University of Technology, Faculty of Civil Engineering, Department of Mathematics and Descriptive Geometry, Bratislava, 2002), 49–57.Google Scholar
[17]Slodička, M., “A robust and efficient linearization scheme for doubly nonlinear and degenerate parabolic problems arising in flow in porous media”, SIAM J. Sci. Comput. 23 (2002) 1593–1614.Google Scholar