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

• Published in: SIAM journal on numerical analysis Vol. 36; no. 3; pp. 890 - 905
• Main Authors: Reinhardt, Hans-Jurgen, Han, Houde, Hao, Dinh Nho
• Format: Journal Article
• Language: English
• Published: Philadelphia, PA Society for Industrial and Applied Mathematics 1999
• Subjects: Applied mathematics; Approximation; Boundary value problems; Cauchy problem; Cauchy problems; Convexity; Difference quotients; Error rates; Exact sciences and technology; Harmonic functions; Laplace equation; Linear transformations; Mathematical analysis; Mathematics; Numerical analysis; Numerical analysis. Scientific computation; Partial differential equations; Partial differential equations, boundary value problems; Regularization methods; Sciences and techniques of general use; Laplace operator; Stability; Error estimation; Sobolev space; Laplace equation; Harmonic function; Cauchy problem; Logarithmic convexity; Boundary value problem; Discretization; Ill posed problem; Regularization; Lagrange identity
• ISSN: 0036-1429; 1095-7170
• DOI: 10.1137/S0036142997316955

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.