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Stability and Regularization of a Discrete Approximation to the Cauchy Problem for Laplace's Equation

Hans-Jurgen Reinhardt, Houde Han and Dinh Nho Hao
SIAM Journal on Numerical Analysis
Vol. 36, No. 3 (1999), pp. 890-905
Stable URL: http://www.jstor.org/stable/2587066
Page Count: 16
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Stability and Regularization of a Discrete Approximation to the Cauchy Problem for Laplace's Equation
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Abstract

The standard five-point difference approximation to the Cauchy problem for Laplace's equation satisfies stability estimates-and hence turns out to be a well-posed problem-when a certain boundedness requirement is fulfilled. The estimates are of logarithmic convexity type. Herewith, a regularization method will be proposed and associated error bounds can be derived. Moreover, the error between the given (continuous) Cauchy problem and the difference approximation obtained via a suitable minimization problem can be estimated by a discretization and a regularization term.

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