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Effects of uncertainties in the domain on the solution of Neumann boundary value problems in two spatial dimensions


Authors: Ivo Babuska and Jan Chleboun
Journal: Math. Comp. 71 (2002), 1339-1370
MSC (2000): Primary 65N99, 65N12, 35J25
DOI: https://doi.org/10.1090/S0025-5718-01-01359-X
Published electronically: June 14, 2001
MathSciNet review: 1933035
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Abstract: An essential part of any boundary value problem is the domain on which the problem is defined. The domain is often given by scanning or another digital image technique with limited resolution. This leads to significant uncertainty in the domain definition. The paper focuses on the impact of the uncertainty in the domain on the Neumann boundary value problem (NBVP). It studies a scalar NBVP defined on a sequence of domains. The sequence is supposed to converge in the set sense to a limit domain. Then the respective sequence of NBVP solutions is examined. First, it is shown that the classical variational formulation is not suitable for this type of problem as even a simple NBVP on a disk approximated by a pixel domain differs much from the solution on the original disk with smooth boundary. A new definition of the NBVP is introduced to avoid this difficulty by means of reformulated natural boundary conditions. Then the convergence of solutions of the NBVP is demonstrated. The uniqueness of the limit solution, however, depends on the stability property of the limit domain. Finally, estimates of the difference between two NBVP solutions on two different but close domains are given.


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Additional Information

Ivo Babuska
Affiliation: The University of Texas at Austin, TICAM, Austin, Texas 78713

Jan Chleboun
Affiliation: Mathematical Institute, Academy of Sciences of the Czech Republic, Žitná 25, 115 67 Prague 1, Czech Republic
Email: chleb@math.cas.cz

DOI: https://doi.org/10.1090/S0025-5718-01-01359-X
Keywords: Elliptic equation, Neumann boundary condition, uncertainty
Received by editor(s): August 5, 1999
Received by editor(s) in revised form: October 13, 2000
Published electronically: June 14, 2001
Additional Notes: The research of the first author was funded partially by the National Science Foundation under the grant NSF–Czech Rep. INT-9724783 and NSF DMS-9802367.
Partial support for the second author, coming from the Ministry of Education of the Czech Republic through grant ME..148(1998) as well as from the Grant Agency of the Czech Republic through grant 201/98/0528, is appreciated.
Article copyright: © Copyright 2001 American Mathematical Society