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

ESAIM: Mathematical Modelling and Numerical Analysis (2010)

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

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topSlodička, 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- 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.
- 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.
- 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.
- 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).
- 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).
- L.C. Evans, Partial differential equations, Graduate Studies in Mathematics19, American Mathematical Society (1998).
- A. Friedman, Variational principles and free-boundary problems. Wiley, New York (1982).
- H. Gerke, U. Hornung, Y. Kelanemer, M. Slodicka and S. Schumacher, Optimal Control of Soil Venting: Mathematical Modeling and Applications, ISNM127, Birkhäuser, Basel (1999).
- D. Gilbarg and N.S. Trudinger, Elliptic Partial Differential Equations of Second Order. Springer, Berlin, Heidelberg (1983).
- 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.
- J. Kacur, Solution to strongly nonlinear parabolic problems by a linear approximation scheme. IMA J. Numer. Anal.19 (1999) 119-145.
- C.V. Pao, Nonlinear parabolic and elliptic equations. Plenum Press, New York (1992).
- 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.
- 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.
- M. Slodicka and H. De Schepper, On an inverse problem of pressure recovery arising from soil venting facilities. Appl. Math. Comput. (to appear).
- M. Slodicka and H. De Schepper, A nonlinear boundary value problem containing nonstandard boundary conditions. Appl. Math. Comput. (to appear).
- 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.
- 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.

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