Solving Inverse PDE by the Finite Element Method

Abstract

In this Master thesis project solving inverse PDE by the finite element method. An optimal control problems subjected to PDE constraint with boundary conditions is given. Construct the variational form then construct Lagrangian, which defined over whole space. Lagrangian function is function of three variables which is defined on whole space, evaluate their partial derivatives, these set of equations are the stationary point equations. Solve these stationary point equations individual, combine and into one equation by finite element method. The error, convergence rate, objective function value, error of objective function is also computed. To calculate finite element solution used FEniCS with Python. Construct the programs in Python and run in FEniCS. Following things are discussed in the following chapters: Chapter 1. Discussion of optimal contol problem with PDE’s constraints. Chapter 2. Discussion of the variational formulation and Lagrangian function. Chapter 3. In this chapter discussion how to solve equations for stationary point by finite element method. To solve equations for stationary point used Python computer program language with FEniCS. FEniCS can be programmed in Python. FEniCS solves partial differential equations, this project used FEniCS to finite element equations by finite element method. In this project a quadratic objective function subjected to linear elliptical partial differential equation with Neumann boundary condition is known, construct the variational form, Lagrangian function which is defined over whole space, taken partial derivatives of this Lagrangian function which gives set of equations are called stationary point equations, write stationary point equations as finite element solution then these equations are called finite element equations. The stationary point equations are used to find the exact solution whereas finite element equations are used to calculate finite element solution. Finally solve these finite element equations as one equation by finite element method, use programming tool FEniCS with Python. The error analysis, convergence rate, objective function value and error of objective function are also computed. The matrix form of stationary point equations are also calculated and show that it is indefinite of saddle point form.