The boundary element solution of the Cauchy steady heat conduction problem in an anisotropic medium

In this paper the iterative algorithm proposed by Kozlov et al. for the Cauchy problem for the Laplace equation is extended in order to solve the Cauchy steady-state heat conduction problem in an anisotropic medium. The iterative algorithm is numerically implemented using the boundary element method (BEM). The convergence and the stability of the numerical method, as well as various types of accuracy, convergence and stopping criteria, are investigated. The numerical results obtained con rm that provided an appropriate stopping regularization criterion is imposed, then the iterative BEM produces a convergent and stable numerical solution with respect to increasing the number of boundary elements and decreasing the amount of noise added into the input data. An e cient stopping regularization criterion to cease the iterative process is proposed and the rate of convergence of the algorithm is improved by using various relaxation procedures between iterations. A new concept of a variable relaxation factor is proposed. Analytical formulae for the coe cients of the matrices resulting from the direct application of the BEM in an anisotropic medium are also presented.