Numerical solution of a Cauchy problem for the Laplace equation
2001 (English)In: Inverse Problems, ISSN 0266-5611, E-ISSN 1361-6420, Vol. 17, no 4, 839-853 p.Article in journal (Refereed) Published
We consider a two-dimensional steady state heat conduction problem. The Laplace equation is valid in a domain with a hole. Temperature and heat-flux data are specified on the outer boundary, and we wish to compute the temperature on the inner boundary. 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. We 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. We propose 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.
Place, publisher, year, edition, pages
2001. Vol. 17, no 4, 839-853 p.
Engineering and Technology
IdentifiersURN: urn:nbn:se:liu:diva-47296DOI: 10.1088/0266-5611/17/4/316OAI: oai:DiVA.org:liu-47296DiVA: diva2:268192