×

zbMATH — the first resource for mathematics

Recovery of an unknown flux in parabolic problems with nonstandard boundary conditions: error estimates. (English) Zbl 1099.65081
In the paper the second order semilinear parabolic initial boundary value problem in a bounded domain \(\Omega \subset \mathbb R^N\) is studied. On nonadjacent parts \(\Gamma _{\text{non}}\) and \(\Gamma _{\text{Dir}}\) of the boundary \(\partial \Omega \) author imposes nonlocal and Dirichlet boundary conditions. On the remaining part \(\Gamma _{\text{Neu}}\) of \(\partial \Omega \), a Robin type boundary condition is set up.
For the construction of a weak solution the Rothe method of time discretization is used. The a-priori estimates obtained by the choice of suitable test functions in the weak formulation of approximated problem are then used in the proof of convergence of the scheme.
In the last section of the paper the error estimates for time discretization are computed and in two examples the exact solutions are compared with the numerical approximations.

MSC:
65M32 Numerical methods for inverse problems for initial value and initial-boundary value problems involving PDEs
35K20 Initial-boundary value problems for second-order parabolic equations
35B30 Dependence of solutions to PDEs on initial and/or boundary data and/or on parameters of PDEs
35K55 Nonlinear parabolic equations
65M15 Error bounds for initial value and initial-boundary value problems involving PDEs
PDF BibTeX XML Cite
Full Text: DOI
References:
[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. · Zbl 0852.35076
[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. · Zbl 0567.65078
[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. · Zbl 0819.76063
[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.
[5] A. Friedman: Partial Differential Equations. Robert E. Krieger Publishing Company, Hungtinton, New York, 1976.
[6] A. Friedman: Variational Principles and Free-Boundary Problems. Wiley, New York, 1982. · Zbl 0564.49002
[7] J. Kačur: Method of Rothe in Evolution Equations. Teubner Texte zur Mathematik Vol. 80. Teubner, Leipzig, 1985. · Zbl 0582.65084
[8] J. Nečas: Les méthodes directes en théorie des équations elliptiques. Academia, Prague, 1967. · Zbl 1225.35003
[9] C. V. Pao: Nonlinear Parabolic and Elliptic Equations. Plenum Press, New York, 1992. · Zbl 0777.35001
[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. <a href=”http://dx.doi.org/10.1002/(SICI)1097-0363(19960315)22:53.0.CO;2-Y” target=”_blank”>DOI 10.1002/(SICI)1097-0363(19960315)22:53.0.CO;2-Y |
[11] K. Rektorys: The Method of Discretization in Time and Partial Differential Equations. Reidel Publishing Company, Dordrecht-Boston-London, 1982. · Zbl 0522.65059
[12] M. Slodička: A monotone linear approximation of a nonlinear elliptic problem with a non-standard boundary condition. 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. · Zbl 1019.35032
[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
[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. · Zbl 1033.35145
[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.
[16] R. Van Keer, M. Slodička: Numerical modelling for the recovery of an unknown flux in semilinear parabolic problems with nonstandard boundary conditions. Proceedings European Congress on Computational Methods in Applied Sciences and Engineering, Barcelona, E. Onate, G. Bugeda and B. Suárez (eds.), Barcelona, 2000.
[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. Numerical Analysis and Its Applications, L. Vulkov, J. Wasniewski and P. Yalamov (eds.), Springer, 2001, pp. 467-474. · Zbl 0978.65089
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.