On the alternating method and boundary-domain integrals for elliptic Cauchy problems

Abstract

A numerical implementation of the alternating iterative method is presented for the Cauchy problem of the conductivity equation in planar annular domains. The conductivity is space dependent as opposed to earlier works, where it is assumed constant. In the constant conductivity case, the mixed problems in the iterations are solved efficiently using boundary integrals. Following on to this, for a space dependent conductivity, it is outlined how to solve these mixed problems using a recent boundary-domain integral equations approach involving the parametrix. For a mixed problem the solution is written as a combination of boundary integrals and a domain integral, with densities to be determined. The densities needed over the boundary and domain are identified by matching against the given mixed boundary data and by requiring the governing equation to hold in the domain. An efficient Nyström scheme is applied for the discretisation. Numerical results are presented for several domains and conductivities, using exact as well as noisy Cauchy data, showing that a stable solution can be obtained with good accuracy and small computational cost.