posteriori error estimates of FEM

Different real-life objects and technological processes are mathematical problems that need to be solved to achieve optimal control over the object described by elliptic differential equations. The technique of finite element method has become one of the main tools of numerical methods for solving differential equations today and has found wide application in the solution of the optimal control problem of systems described by differential equations.In this book, we describe the finite element method, error estimation method and detailed numerical simulation method to solve the optimal control problem of some semi-linear second order elliptic differential equations in two dimensions.In Chapter 1, finite element posteriori error estimation, convergence and numerical simulation for the source optimal control problem of semi-linear elliptic equations with nonlinear Robin boundary conditions are presented.In Chapter 2, we evaluate the finite element posteriori error and the priori error for the source optimal control problem of a semi-linear system of convection-diffusion equations with Dirichlet boundary conditions and describe the numerical simulations.