A Posteriori Error Estimates by FEM for Source Control Problems Governed by a System of Semi-Linear Convection-Diffusion Equations

In this paper, we consider the optimal source control problem of a system of 2-dimensional semi-linear steady convection-diffusion equations. The problem is modelized from temperature and consistency distribution in the gasification processes, so it is described by 2 non-linear elliptic partial differential equations with Dirichlet boundary condition. The problem is a optimal source control problem that controls the source term necessary to approximate the temperature to a proper target function. First, we derived the optimal condition. 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 by using optimal condition of original and approximate problem. And we also evaluated the upper estimate of a posteriori error by finite element method (FEM). We proved the convergence to 0 of a posteriori error indicator (term of the right side of inequality) when division diameter converges to 0. For this, we acquired the lower bound estimation of a posteriori error and proved that a priori error and total variance error converges to 0 when division diameter converges to 0, so that we proved the convergence problem of a posteriori error indicator.