| Title:
|
Recovery of an unknown flux in parabolic problems with nonstandard boundary conditions: Error estimates (English) |
| Author:
|
Slodička, Marián |
| Language:
|
English |
| Journal:
|
Applications of Mathematics |
| ISSN:
|
0862-7940 (print) |
| ISSN:
|
1572-9109 (online) |
| Volume:
|
48 |
| Issue:
|
1 |
| Year:
|
2003 |
| Pages:
|
49-66 |
| Summary lang:
|
English |
| . |
| Category:
|
math |
| . |
| Summary:
|
In this paper, we consider a 2nd order semilinear parabolic initial boundary value problem (IBVP) on a bounded domain $\Omega \subset \mathbb{R}^N$, with nonstandard boundary conditions (BCs). More precisely, at some part of the boundary we impose a Neumann BC containing an unknown additive space-constant $\alpha (t)$, accompanied with a nonlocal (integral) Dirichlet side condition. We design a numerical scheme for the approximation of a weak solution to the IBVP and derive error estimates for the approximation of the solution $u$ and also of the unknown function $\alpha $. (English) |
| Keyword:
|
nonlocal boundary condition |
| Keyword:
|
parameter identification |
| Keyword:
|
parabolic IBVP |
| MSC:
|
35B30 |
| MSC:
|
35K20 |
| MSC:
|
35K55 |
| MSC:
|
65M15 |
| MSC:
|
65M32 |
| idZBL:
|
Zbl 1099.65081 |
| idMR:
|
MR1954503 |
| DOI:
|
10.1023/A:1022954920827 |
| . |
| Date available:
|
2009-09-22T18:12:13Z |
| Last updated:
|
2020-07-02 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/134516 |
| . |
| Reference:
|
[1] D. Andreucci, R. Gianni: Global existence and blow up in a parabolic problem with nonlocal dynamical boundary conditions.Adv. Differential Equations 1 (1996), 729–752. MR 1392003 |
| Reference:
|
[2] D. N. Arnold, F. Brezzi: Mixed and nonconforming finite element methods: implementation, postprocessing and error estimates.RAIRO Modél. Math. Anal. Numér. 19 (1985), 7–32. MR 0813687, 10.1051/m2an/1985190100071 |
| Reference:
|
[3] J. H. Bramble, P. Lee: On variational formulations for the Stokes equations with nonstandard boundary conditions.RAIRO Modél. Math. Anal. Numér. 28 (1994), 903–919. MR 1309419, 10.1051/m2an/1994280709031 |
| Reference:
|
[4] H. De Schepper, M. Slodička: Recovery of the boundary data for a linear second order elliptic problem with a nonlocal boundary condition.ANZIAM Journal (C) 42 (2000), 518–535. MR 1810647, 10.21914/anziamj.v42i0.611 |
| Reference:
|
[5] A. Friedman: Partial Differential Equations.Robert E. Krieger Publishing Company, Hungtinton, New York, 1976. MR 0454266 |
| Reference:
|
[6] A. Friedman: Variational Principles and Free-Boundary Problems.Wiley, New York, 1982. Zbl 0564.49002, MR 0679313 |
| Reference:
|
[7] J. Kačur: Method of Rothe in Evolution Equations. Teubner Texte zur Mathematik Vol. 80.Teubner, Leipzig, 1985. MR 0834176 |
| Reference:
|
[8] J. Nečas: Les méthodes directes en théorie des équations elliptiques.Academia, Prague, 1967. MR 0227584 |
| Reference:
|
[9] C. V. Pao: Nonlinear Parabolic and Elliptic Equations.Plenum Press, New York, 1992. Zbl 0777.35001, MR 1212084 |
| Reference:
|
[10] J. Heywood, 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. MR 1380844, 10.1002/(SICI)1097-0363(19960315)22:5<325::AID-FLD307>3.0.CO;2-Y |
| Reference:
|
[11] K. Rektorys: The Method of Discretization in Time and Partial Differential Equations.Reidel Publishing Company, Dordrecht-Boston-London, 1982. Zbl 0522.65059, MR 0689712 |
| Reference:
|
[12] 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.), Slovak University of Technology, Faculty of Civil Engineering, Department of Mathematics and Descriptive Geometry, Bratislava, 2000, pp. 47–57. |
| Reference:
|
[13] M. Slodička: 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. Zbl 0997.65124, MR 1862875, 10.1051/m2an:2001132 |
| Reference:
|
[14] M. Slodička and H. De Schepper: On an inverse problem of pressure recovery arising from soil venting facilities.Appl. Math. Comput. 129 (2002), 469–480. MR 1905411, 10.1016/S0096-3003(01)00057-1 |
| Reference:
|
[15] 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. MR 1675223, 10.1007/BF02905911 |
| Reference:
|
[16] R. Van Keer, M. Slodička: Numerical modelling for the recovery of an unknown flux in semilinear parabolic problems with nonstandard boundary conditions.In: Proceedings European Congress on Computational Methods in Applied Sciences and Engineering, Barcelona, E. Onate, G. Bugeda and B. Suárez (eds.), Barcelona, 2000. |
| Reference:
|
[17] R. Van Keer, M. Slodička: Numerical techniques for the recovery of an unknown Dirichlet data function in semilinear parabolic problems with nonstandard boundary conditions.In: Numerical Analysis and Its Applications, L. Vulkov, J. Wasniewski and P. Yalamov (eds.), Springer, 2001, pp. 467–474. MR 1938440 |
| . |
