Abstract
In this paper, we consider an ill-posed boundary value problem for the equation \(\operatorname{div}A\nabla u+f=0\), which is closely connected with a problem of reconstruction of an unknown boundary condition. This problem can be reformulated as an unconstrained minimization problem for a convex nonnegative functional depending on the pair of variables (v,q), which approximate the desired solution and its flux, respectively. The functional vanishes if and only if v and q coincide with the exact solution of the problem (if the latter solution exists) and its flux, respectively. Moreover, we prove that if the functional is lesser than a small positive number ε, then ε-neighborhood of (v,q) contains the exact solution of the direct boundary value problem with mixed boundary conditions, which are traces on the boundary ε-close to the Cauchy conditions imposed. Advanced forms of the functional convenient for numerical computations are discussed.
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Repin, S., Rossi, T. (2013). On Quantitative Analysis of an Ill-Posed Elliptic Problem with Cauchy Boundary Conditions. In: Repin, S., Tiihonen, T., Tuovinen, T. (eds) Numerical Methods for Differential Equations, Optimization, and Technological Problems. Computational Methods in Applied Sciences, vol 27. Springer, Dordrecht. https://doi.org/10.1007/978-94-007-5288-7_7
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DOI: https://doi.org/10.1007/978-94-007-5288-7_7
Publisher Name: Springer, Dordrecht
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