• inverse natural convection problem;
  • adjoint variable method;
  • direct differentiation method;
  • conjugate gradient method;
  • consistent splitting method;
  • equal-order/mixed finite elements


Several numerical algorithms for solving inverse natural convection problems are revisited and studied. Our aim is to identify the unknown strength of a time-varying heat source via a set of coupled nonlinear partial differential equations obtained by the so-called finite element consistent splitting scheme (CSS) in order to get a good approximation of the unknown heat source from both the measured data and model results, by minimizing a functional that measures discrepancies between model and measured data. Viewed as an optimization problem, the solutions are obtained by means of the conjugate gradient method. A second-order CSS in time involving the direct problem, the adjoint problem, the sensitivity problem and a system of sensitivity functions is used in order to enhance the numerical accuracy obtained for the unknown heat source function. A spatial discretization of all field equations is implemented using equal-order and mixed finite element methods. Numerical experiments validate the proposed optimization algorithms that are in good agreement with the existing results. Copyright © 2010 John Wiley & Sons, Ltd.