Adjoint method for inverse boundary value problem of heat conduction

An inverse boundary value problem for the equation of steady heat conduction is considered. Temperature and heat flux are prescribed on respective part of the boundary, while there is the second part of the boundary where no measurement is done. There is the third part of the boundary where the Newton cooling is prescribed. This problem of finding unknown values along the whole boundary is reformulated in terms of the variational problem, which is then recast into primary and adjoint boundary value problems of the Laplace equation in the conventional forms. A direct method for numerical solution of the boundary value problems using the boundary element method is presented.