Unique discrete harmonic continuation and data completion problems using the fading regularization method

A data completion method is proposed for solving Cauchy problems for which the number of data is less than the number of unknowns. This method is presented on the Cauchy problem for the Laplace equation in 2D situations. The idea is to search the solution in a space of discrete harmonic functions for which the existence and the uniqueness of a discrete harmonic continuation are guaranteed. Many numerical simulations using the finite element method highlight the efficiency, accuracy, and stability of this method even when data are noisy.

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