A Priori and a Posteriori Error Estimates of Finite Element Method for Source Control Problems Governed by a System of Quasi-linear Elliptic Equations

In this paper, we consider the optimal source control of a 2-dimensional steady-state thermistor. The problem is described by a system of two nonlinear elliptic partial differential equations with appropriate boundary conditions which model the coupling of the thermistor to its surroundings. The heat source is Joule heat due to variable resistance. The problem is a source optimal control problem that controls the source term necessary to approximate the temperature to a proper target function. First, we derive the optimality condition of the problem. Based on setting the approximation problem of a given control problem in a first order polynomial finite element function space and deriving the optimality condition of the approximation problem, we evaluated a priori error between the optimal control, the optimal state, the conjugate state and its finite element approximation functions. Then, we evaluate the upper bound of a posteriori error estimates that are currently available for error estimation. For a posteriori error estimates, it is necessary to find the convergence of the error indicator. In this paper, we prove the convergence of a posteriori error indicator by obtaining a lower bound estimate of a posteriori error and finding that the total variance error goes to zero. And, we propose a gradient algorithm to find the optimal control and provide a condition for this algorithm to converge. The validity is also demonstrated by adaptive numerical simulations with a detailed problem. The computational results are obtained on three adaptive meshes and the graphs of the finite element solutions are presented.