Finite Element Method For Optimal Boundary Control Problem Governed By Westervelt Equation

Westervelt equation is a basic mathematical model in the field of nonlinear acoustic wave equation.It plays a important role in several medical and in-dustrial applications,such as,lithotripsy,thermotherapy,ultrasound cleaning or welding,and sonochemistry and so on.In these practical applications,the a-coustic pressure is excited by some piezoelectric transducers.On one part of the boundary,the normal acceleration of the piezoelectric transducers stipulates the normal derivative of the acoustic pressure,which leads to use the controllable variable on Neumann boundary condition to simulate the effects of piezoelectric transducers on the pressure.Then,optimal boundary control problem governed by Westervelt equation is formed.This thesis considers finite element method for optimal boundary control problem governed by Westervelt equation,which consists of three chapters.Chapter 1 introduces the model,the backgrounds,some related research re-sults and main difficulties of optimal boundary control problems governed by Westervelt equation.Some main contents and conclusions of the thesis are sum-marized.Chapter 2 considers finite element method for optimal boundary control prob-lem governed by linearized Westervelt equation.First,we present the description of this problem and deduce the adjoint state equation and the optimal condi-tion by the theory of optimal control problem.Then,the objective functional in discrete form is given and the second-order finite element scheme about time is established.Next,the state and co-state variables are discretized by piecewise linear continuous functions and the control variable is approximated by piecewise constants.By use of the analytical skills of a priori error estimates,the optimal L?(0,T;H1(?))-norm error estimates for the state and co-state variables and op-timal L?(0,T;L2(r))-norm error estimates for the control variable are derived,i.e.,O(hU+ h+(?t)2).Finally,we present a numerical example and numerical experiments to verify the correctness and the validity of the theoretical results.Chapter 3 discusses optimal boundary control problem governed by Westervelt equation.First,we explain this problem is in fact of state constrained.Then,we quote the penalty method and point out that the theoretical analysis of finite element method for this problem is very difficult due to the complexity of the penalized objective functional and the optimal system.Finally,based on the finite element scheme for the linearized case in Chapter 2,we propose an iterative numerical algorithm.A numerical example and numerical experiments are carried out to obtain good numerical results.The effective calculation of the optimal boundary control problem by Westervelt equation is preliminarily realized.