FDM-based alternating iterative algorithms for inverse BVPs in 2D steady-state anisotropic heat conduction with heat sources

Abstract

In the framework of steady-state inhomogeneous anisotropic heat conduction with heat sources, we study the convergent and stable numerical reconstruction of the missing boundary conditions (temperatures and normal heat fluxes) on an inaccessible boundary and the thermal field in a 2D solid from the knowledge of over-prescribed data on an accessible portion of the boundary and temperature data on the remaining boundary. This inverse boundary value problem is approached by employing the alternating iterative algorithms of Kozlov et al. (1991), in conjunction with the standard finite-difference method and a novel modified one, for both exact and perturbed data. For noisy data available on the over-prescribed boundary, the numerical solution is retrieved by using three regularising/stabilising stopping criteria. Numerical results are presented for 2D simply and doubly connected domains bounded by a (piecewise) smooth curve, as well as exact and noisy boundary data, whilst appropriate errors in the fractional Sobolev spaces corresponding to the boundary temperatures and normal heat fluxes, respectively, are computed efficiently.