On the numerical solution of a Cauchy problem for the Laplace equation via a direct integral equation approach

We investigate the problem of determining the stationary temperature
field on an inclusion from given Cauchy data on an accessible exterior
boundary. On this accessible part the temperature (or the heat flux) is known,
and, additionally, on a portion of this exterior boundary the heat flux (or temperature) is also given. We propose a direct boundary integral approach in
combination with Tikhonov regularization for the stable determination of the
temperature and flux on the inclusion. To determine these quantities on the
inclusion, boundary integral equations are derived using Green’s functions, and
properties of these equations are shown in an L2-setting. An effective way of
discretizing these boundary integral equations based on the Nystr¨om method
and trigonometric approximations, is outlined. Numerical examples are included, both with exact and noisy data, showing that accurate approximations
can be obtained with small computational effort, and the accuracy is increasing
with the length of the portion of the boundary where the additionally data is
given.