Numerical solution of a Cauchy problem for the Laplace equation (Q2747526)

A two-dimensional steady state heat conduction problem is considered. The Laplace equation is valid in a domain with a hole. Temperature and heat-flux data are specified on the outer boundary, and the temperature on the inner boundary is to be computed. This Cauchy problem is ill-posed, i. e. the solution does not depend continuously on the boundary data, and small errors in the data can destroy the numerical solution. NEWLINENEWLINENEWLINEThe authors consider two numerical methods for solving this problem. A standard approach is to discretize the differential equation by finite differences, and use Tikhonov regularization on the discrete problem, which leads to a large sparse least squares problem. It is proposed to use a conformal mapping that maps the region onto an annulus, where the equivalent problem is solved using a technique based on the fast Fourier transform. The ill-posedness is dealt with by filtering away high frequencies in the solution. Numerical results using both methods are given.