Finite element methods for parameter identification problem of linear and nonlinear steady-state diffusion equations

Abstract

We study a parameter identification problem for the steady state diffusion equations. In this thesis, we transform this identification problem into a minimization problem by considering an appropriate cost functional and propose a finite element method for the identification of the parameter for the linear and nonlinear partial differential equation. The cost functional involves the classical output least square term, a term approximating the derivative of the piezometric head 𝑢(𝑥), an equation error term plus some regularization terms, which happen to be a norm or a semi-norm of the variables in the cost functional in an appropriate Sobolev space. The existence and uniqueness of the minimizer for the cost functional is proved. Error estimates in a weighted 𝐻⁻¹-norm, 𝐿²-norm and 𝐿¹-norm for the numerical solution are derived. Numerical examples will be given to show features of this numerical method.