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

Marian Slodička

ESAIM: Mathematical Modelling and Numerical Analysis (2010)

  • Volume: 35, Issue: 4, page 691-711
  • ISSN: 0764-583X

Abstract

top
We consider a nonlinear second order elliptic boundary value problem (BVP) in a bounded domain Ω N 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 Γn. 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 L2(Ω),H1(Ω) and L∞(Ω) spaces.

How to cite

top

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 35.4 (2010): 691-711. <http://eudml.org/doc/197607>.

@article{Slodička2010,
abstract = { We consider a nonlinear second order elliptic boundary value problem (BVP) in a bounded domain $\Omega\subset \{\mathbb R\}^N$ 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 Γn. 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 L2(Ω),H1(Ω) and L∞(Ω) spaces. },
author = {Slodička, Marian},
journal = {ESAIM: Mathematical Modelling and Numerical Analysis},
keywords = {Nonlinear elliptic BVP; error estimates; nonstandard boundary condition; linearization.; nonlinear elliptic boundar value problem; nonstandard boundary conditions; linearization; convergence; numerical examples},
language = {eng},
month = {3},
number = {4},
pages = {691-711},
publisher = {EDP Sciences},
title = {Error estimates of an efficient linearization scheme for a nonlinear elliptic problem with a nonlocal boundary condition},
url = {http://eudml.org/doc/197607},
volume = {35},
year = {2010},
}

TY - JOUR
AU - Slodička, Marian
TI - Error estimates of an efficient linearization scheme for a nonlinear elliptic problem with a nonlocal boundary condition
JO - ESAIM: Mathematical Modelling and Numerical Analysis
DA - 2010/3//
PB - EDP Sciences
VL - 35
IS - 4
SP - 691
EP - 711
AB - We consider a nonlinear second order elliptic boundary value problem (BVP) in a bounded domain $\Omega\subset {\mathbb R}^N$ 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 Γn. 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 L2(Ω),H1(Ω) and L∞(Ω) spaces.
LA - eng
KW - Nonlinear elliptic BVP; error estimates; nonstandard boundary condition; linearization.; nonlinear elliptic boundar value problem; nonstandard boundary conditions; linearization; convergence; numerical examples
UR - http://eudml.org/doc/197607
ER -

References

top
  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.  Zbl0852.35076
  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.  Zbl0567.65078
  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.  Zbl0819.76063
  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).  
  5. H. De Schepper and M. Slodicka, 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).  
  6. L.C. Evans, Partial differential equations, Graduate Studies in Mathematics19, American Mathematical Society (1998).  Zbl0902.35002
  7. A. Friedman, Variational principles and free-boundary problems. Wiley, New York (1982).  Zbl0564.49002
  8. H. Gerke, U. Hornung, Y. Kelanemer, M. Slodicka and S. Schumacher, Optimal Control of Soil Venting: Mathematical Modeling and Applications, ISNM127, Birkhäuser, Basel (1999).  Zbl0919.73001
  9. D. Gilbarg and N.S. Trudinger, Elliptic Partial Differential Equations of Second Order. Springer, Berlin, Heidelberg (1983).  Zbl0562.35001
  10. W. Jäger and J. Kacur, Solution of doubly nonlinear and degenerate parabolic problems by relaxation schemes. RAIRO Modél. Math. Anal. Numér.29 (1995) 605-627.  Zbl0837.65103
  11. J. Kacur, Solution to strongly nonlinear parabolic problems by a linear approximation scheme. IMA J. Numer. Anal.19 (1999) 119-145.  Zbl0946.65145
  12. C.V. Pao, Nonlinear parabolic and elliptic equations. Plenum Press, New York (1992).  Zbl0777.35001
  13. R. Rannacher and S. Turek, Artificial boundaries and flux and pressure conditions for the incompressible Navier-Stokes equations. Internat. J. Numer. Methods Fluids22 (1996) 325-352.  Zbl0863.76016
  14. M. Slodicka, A monotone linear approximation of a nonlinear elliptic problem with a non-standard boundary condition, in Algoritmy 2000, A. Handlovicová, M. Komorníková, K. Mikula and D. Sevcovic, Eds., Bratislava (2000) 47-57.  Zbl1019.35032
  15. M. Slodicka and H. De Schepper, On an inverse problem of pressure recovery arising from soil venting facilities. Appl. Math. Comput. (to appear).  Zbl1033.35145
  16. M. Slodicka and H. De Schepper, A nonlinear boundary value problem containing nonstandard boundary conditions. Appl. Math. Comput. (to appear).  Zbl1135.35341
  17. M. Slodicka and R. Van Keer, A nonlinear elliptic equation with a nonlocal boundary condition solved by linearization. Internat. J. Appl. Math.6 (2001) 1-22.  Zbl1030.35082
  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.  

NotesEmbed ?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.