Numerical Reconstruction of Heat Fluxes

This paper studies the reconstruction of heat fluxes on an inner boundary of a heat conductive system when the measurement of temperature in a small subregion near the outer boundary of the physical domain is available. We will first consider two different regularization formulations for this severely ill-posed inverse problem and justify their well-posedness; then we will propose two fully discrete finite element methods to approximate the resultant nonlinear minimization problems. The existence and uniqueness of the discrete minimizers and convergence of the finite element solution are rigorously demonstrated. A conjugate gradient method is formulated to solve the nonlinear finite element optimization problems. Numerical experiments are given to demonstrate the stability and effectiveness of the proposed reconstruction methods.