  In the present paper we investigate the inverse problem of identifying
simultaneously the diffusion matrix, source term and boundary condition as well
as the state in the Neumann boundary value problem for an elliptic partial
differential equation (PDE) from a measurement data, which is weaker than
required of the exact state. A variational method based on energy functions
with Tikhonov regularization is here proposed to treat the identification
problem. We discretize the PDE with the finite element method and prove the
convergence as well as analyse error bounds of this approach.
