Numerical solution of an inverse steady state heat conduction problem

摘要

We consider a two dimensional inverse steady state heat conduction problem. The Laplace equation is valid in a domain with a hole, and temperature and heat-flux data are specified on the outer boundary. The 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. We consider two numerical methods for solving this problem. By discretizing the differential equation using finite differences, and using Tikhonov regularization on the discrete problem, we get a large sparse least squares problem. Alternatively we can use a conformal mapping, and solve an equivalent problem on an annulus, using a technique based on the Fast Fourier transform. Numerical results using both methods are given.