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It is well known that Cauchy problem for Laplace equations is an ill-posedproblem in Hadamard's sense. Small deviations in Cauchy data may lead to largeerrors in the solutions. It is observed that if a bound is imposed on thesolution, there exists a conditional stability estimate. This gives areasonable way to construct stable algorithms. However, it is impossible tohave good results at all points in the domain. Although numerical methods forCauchy problems for Laplace equations have been widely studied for quite a longtime, there are still some unclear points, for example, how to evaluate thenumerical solutions, which means whether we can approximate the Cauchy datawell and keep the bound of the solution, and at which points the numericalresults are reliable? In this paper, we will prove the conditional stabilityestimate which is quantitatively related to harmonic measures. The harmonicmeasure can be used as an indicate function to pointwisely evaluate thenumerical result, which further enables us to find a reliable subdomain wherethe local convergence rate is higher than a certain order.

Harmonic Measures and Numerical Computation of Cauchy Problems for Laplace Equations