# Article

 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 .

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