Sensitivity analysis of a parabolic-elliptic problem
Author:
Bastian Gebauer
Journal:
Quart. Appl. Math. 65 (2007), 591-604
MSC (2000):
Primary 35K65, 35B40, 35M10
DOI:
https://doi.org/10.1090/S0033-569X-07-01072-4
Published electronically:
August 2, 2007
MathSciNet review:
2354889
Full-text PDF Free Access
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Additional Information
Abstract: We consider the heat flux through a domain with subregions in which the thermal capacity approaches zero. In these subregions the parabolic heat equation degenerates to an elliptic one. We show the well-posedness of such parabolic-elliptic differential equations for general non-negative $L^\infty$-capacities and study the continuity of the solutions with respect to the capacity, thus giving a rigorous justification for modeling a small thermal capacity by setting it to zero. We also characterize weak directional derivatives of the temperature with respect to capacity as solutions of related parabolic-elliptic problems.
References
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References
- H. Ammari, A. Buffa, J.-C. Nédélec, A justification of eddy currents model for the Maxwell equations, SIAM J. Appl. Math. 60 (2000), 1805–1823. MR 1761772 (2001g:78003)
- M. Costabel, Boundary integral operators for the heat equation, Integral Equations Oper. Theory 13 (1990), 488–552. MR 1058085 (91j:35119)
- M. Costabel, V. J. Ervin, E. P. Stephan, Symmetric coupling of finite elements and boundary elements for a parabolic-elliptic interface problem, Quart. Appl. Math. 48 (1990), 265–279. MR 1052136 (92e:65145)
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- M. Renardy, R. C. Rogers, An introduction to partial differential equations, Texts Appl. Math., vol. 13, Springer-Verlag, New York, 1993. MR 1211418 (94c:35001)
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Additional Information
Bastian Gebauer
Affiliation:
Johann Radon Institute for Computational and Applied Mathematics (RICAM), Austrian Academy of Sciences, Altenbergerstr. 69, 4040 Linz, Austria
Email:
bastian.gebauer@ricam.oeaw.ac.at
Keywords:
Parabolic-elliptic equation,
degenerate parabolic equation,
asymptotic behavior,
sensitivity analysis
Received by editor(s):
March 15, 2007
Published electronically:
August 2, 2007
Article copyright:
© Copyright 2007
Brown University
The copyright for this article reverts to public domain 28 years after publication.