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

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.