On the use of an integral equation approach for the numerical solution of a Cauchy problem for Laplace equation in a doubly connected planar domain

We consider the Cauchy problem for the Laplace equation, i.e. the reconstruction of a harmonic function from knowledge of the value of the function and its normal derivative given on a part of the boundary of the solution domain. The solution domain considered is a bounded smooth doubly connected planar domain bounded by two simple disjoint closed curves. Since the Cauchy problem is ill-posed, as a regularizing method we generalize the novel direct integral equation approach in [1], originally proposed for a circular outer boundary curve, to a more general simply connected domain. The solution is represented as a sum of two single-layer potentials defined on each of the two boundary curves and in which both densities are unknown. To identify these densities, the representation is matched with the given Cauchy data to generate a system to solve for the densities. It is shown that the operator corresponding to this system is injective and has dense range, thus Tikhonov regularization is applied to solve it in a stable way. For the discretisation, the Nystrom method is employed generating a linear system to solve, and via Tikhonov regularization a stable discrete approximation to these densities are obtained. Using these one can then find an approximation to the solution of the Cauchy problem. A numerical example is included and we compare with other regularizing methods as well (implemented via integral methods). These results show that the proposed direct method gives accurate reconstructions with little computational effort (the computational time is of order of seconds). Moreover, the obtained approximation can be used as an initial guess in more involved regularizing methods to further improve the accuracy.