Analysis and Optimal Control of Phase-Field Transition System: Fractional Steps Methods

The aim of this Chapter is to provide some basic results that we will use frequently in the chapters that follow. Most results presented here are given without proof. We refer the reader to the appropriate monographs or textbooks where details can be found.

In this chapter we will deal, on one hand, with the study of the existence, uniqueness, regularity and estimates of the solution of phase-field transition system, and on the other hand, we will analyze the convergence of some approximating schemes of fractional steps type, associated to this nonlinear system.

system. In the first Section the existence, uniqueness and regularity of the solution of the phase-field transition system subject to the non-homogeneous Cauchy-Neumann boundary conditions (Theorem 2.1) are studied. Such kind of conditions allow the phase-field system (Caginalp’s model) to be considered as a model of heat transfer from the surface of product to the environment, when we assume that heat is extracted by convection and conduction in the continuous casting process.....

Some types of boundary optimal control problems governed by the nonlinear phase-field transition system, are introduced and analyzed in this Chapter.

The aim of Section 3.1 is to prove (for later use) a priori estimates in L2([0, T];H2( )) for unknown u, ' in phase-field system, in the presence of following boundary conditions (see (30), (300) in Preface):....

Two topics will be covered in this Chapter:

• numerical approximation of the solution of nonlinear phase-field transition system;

• numerical approximation of boundary optimal control, governed by phase-field transition system. For both subjects we consider the phase-field system with non-homogeneous Cauchy-Neumann boundary conditions:....

In this Chapter we give a description of the programs implementing the conceptual algorithms introduced in Chapter 4 to approximate the solution of nonlinear phase-field transition system. We also give some numerical results in order to illustrate the description of the programs and to convince readers of their correctness, in terms of syntactic and semantic.

All programs in this Chapter were written in MATLAB and have been endowed with a sufficient amount of comments.

The existence and regularity for the solution of one linear system (auxiliary linear system), having a structure similar with those that appear in the associated approximating schemes to the nonlinear phase-field transition system, it is proved in this Annexe (Theorem A.1). Concerning the methods used in the proof, an essential difference with respect to Chapter 2 consists in that, presently, we make the a priori estimates in L2(Q) in place of Lp(Q). Estimation technique that is used is also different from those used for the rest of the book......

The finite element method (fem for short) is a general method for approximating the solution of boundary value problems for partial differential equations. This method derives from the Ritz (or Gelerkin) method, characteristic for the finite element method being the choice of the finite dimensional space, namely, in the case of fem the finite dimensional space, corresponding to the original space of functions, is the span of a set of finite element basis functions, as we will see in the sequel.