The Methods Of Characteristic Finite Element Of Optimal Control Problems Governed By Convection-Diffusion Equations

The optimal control or design of partial differential equations is a very lively and active mathematical field, and it has been widely studied and applied in the last 30 years. Being as a branch of mathematics, especially applied mathematics, the research of optimal control covers a lot of topics, such as time optimal control, feedback control, flow control, optimal shape design including material design, crystal growth and chemical reaction, optimal control of multi-scale problems, and parameter estimation, etc. We refer the reader to see [55, 71, 72, 80, 81], for example. In the last 5 years, people are interested in the following aspects:Discretization;Fast solver for the large optimization problems;Adaptive FEM;Realistic algorithm for 3-D time-dependent problems;Multi-scale system control.In general, most of the optimal control problems that we are interested in can be symbolically written in the following form: (OCP)s.t.where J is the objective functional;y is the state variable in a state space V ; u is the control variable in a control space U;U_ad is the admissible set of the control (design);A is a suitable operator on V×U_ad.Generally the state equation A(y;u) = 0 can be a PDE, or a variational inequality, etc., and may incorporate state constraints. More details will be given below. Thus the mechanism is to control the state (like temperature or position of a system, usually not easy to control directly) through the control variable, which drives the state through the state equation. Sometimes, the control variable is called design variable, the control space is referred to as design space, and the state variable may be referred to as system variable.Optimal control or design is crucial to many engineering applications. Efficient numerical methods are essential to successful applications of optimal control. Nowadays, the finite element method seems to be the most widely used numerical method in computing optimal control problems, and there are indeed very extensive studies in the standard finite element approximation of various optimal control problems including the a priori/posteriori error estimates and convergence analysis of optimal control problems governed by elliptic or time-dependent equations; see, for example, [4, 9, 31, 35, 48, 52, 60, 61, 62, 63, 65, 70, 71] and the references cited therein. However, it is impossible to give even a very brief review here. Systematic introductions of the finite element method for PDEs and optimal control problems can be found in, for example, [24, 55, 72, 81].Although the finite element approximation of optimal control problems has been extensively studied in the literature. See, some examples, in [35. 39, 52] for linear elliptic equations, in [5] for semilinear elliptic equations, in [60, 64] for nonlinear elliptic equations, in [57, 62] for stokes equations, in [4, 53, 63, 69, 70] for linear parabolic equations and in [11, 25, 87, 89] for other equations. To the best of my remembrance, we are the first one who publish optimal control papers cover the time-dependent convection-diffusion equations, see [32] or the part of results in Chapter 1 for details. We recall that time-dependent convection-diffusion partial differential equations arise in mathematical models of petroleum reservoir simulation, environmental modeling, groundwater contaminant transport, transport problems of heat, solute in moving fluids and many other applications [8, 29]. As we all know that in many applications the Peclet number is very high, so these problems become strong convection dominant and they admit solutions with moving steep fronts and complex structures and present serious mathematical and numerical difficulties. It is well known that the classical standard techniques such as finite difference method or finite element discretizations applied to convection dominated diffusion problems tend to generate numerical solutions with strong and unacceptable nonphysical oscillations when layers are not properly resolved, while standard upwind methods used in such problems often artificially smear moving fronts with excessive numerical dispersion and produce solutions that depend strongly on the orientation of the difference grid relative to the streamlines of flow. To stabilize these phenomena, several well-established techniques have been proposed and analyzed, for example, the streamline diffusion finite element method [44], residual-free bubbles [16], the discontinuous Galerkin method [49], edge-stabilization Galerkin method [17], and the characteristic methods (such as MMOC [27, 30], MMOCAA [26], ELLAM [21, 83], CMFEM [6]) for time-dependent problems.Although the above techniques are deeply studied for the convection-dominated diffusion equations, it is not so straight to use them for the optimal control problems in many cases. To our best knowledge, up to now there are only few published results on optimal control problems governed by steady convection-diffusion equations, see [25] by use of SUPG method; [11] using standard finite element discretizations with stabilization based on local projections (the so called LPS) method; [87] using edge-stabilization method. There are also few published results on these topics for parabolic problems, see, for example [53, 63, 69, 70]. We even do not see any one who has published related papers on these topics governed by unsteady convection-diffusion equations except us. As we all know characteristic(-curve) methods have been developed to perform stable computation, and the procedure of the characteristic method is natural from the physical point of view since it approximates particle movements. Besides, it is attractive from the mathematical point of view since it symmetrizes the problem. Based on these points, my main work during the doctor's degree is centered on the characteristic finite element (CFE) approximation of optimal control problems governed by transient convection-diffusion equations. We discuss two kinds of convection-diffusion equations. For the first class, the velocity field is incompressible and a priori and a posteriori error estimates are obtained. The main work of this class is presented in Chapter 1 and Chapter 2. Then we generalize it to the divergence-form convection-diffusion equations with non-divergence-free velocity field, and another a priori error analysis are given. This work is delivered in Chapter 3. Among many kinds of finite element methods, adaptive finite element methods based on a posteriori error estimates have become a central theme in scientific and engineer computations for their high efficiency. In order to obtain a numerical solution of acceptable accuracy for the optimal control problems with obstacle constraints, the finite element meshes have to be refined according to a mesh refinement scheme. Adaptive finite element approximation is of very importance in improving accuracy and efficiency of the finite element discretizations. It ensures a higher density of nodes in certain area of the given domain, where the solution is more difficult to approximate, using a posteriori error indicator. In this sense, efficiency and reliability of adaptive finite element approximation rely very much on the error indicator used. There exists several concepts including residual and hierarchical type estimators, error estimates that based on local averaging, the so-called goal-oriented dual weighted approach, and functional type error majorants (cf. [4, 7, 82]). The theory and application of adaptive finite element methods for the efficient numerical solution of boundary and initial-boundary value problems for partial differential equations has reached some state of maturity as documented by a series of monographs. However, On the other hand, as far as the development of adaptive finite element schemes for optimal control problems of partial differential equations is concerned, much less work has been done. The goal-oriented dual weighted approach has been applied to unstrained problems in [9]. Residual-type a posteriori error estimators for control constrained problems have been derived and analyzed in [41. 60, 61, 70] and so on. In a constrained control problem, the optimal control and the state variables usually have different regularity, and what is more, the locations of the singularities are very different. In general, the optimal control has only limited regularity (at most in H¹(Ω)). This indicates that the traditional all-in-one mesh strategy for control and state may be inefficient. Therefore, adaptive multi-mesh; that is. separate adaptive meshes which are adjusted according to different error indicators, are often necessary. Particularly, it seems to be efficient to use multi-set adaptive meshes in applying adaptive finite element methods to compute optimal control problems, see [48, 51] for details. However, it is much more complicated to implement adaptive computation for evolution control problems, see [53, 63]. for example.Under the aborative guidance of Professor Hongxing Rui, the author has finished this dissertation consisting of some work on optimal control problems governed by transient convection-diffusion equations. Based on the idea of optimal control problem governed by parabolic equations proposed by Professor Wenbin Liu, et al, combined with the well-known characteristic finite element methods, two different class of time-dependent convection-diffusion equations for optimal control problems are presented. We analyze the convergence of each procedure and derive some a priori and a posteriori error estimates. Both theoretical analysis and numerical experiments show that these algorithms proposed in this dissertation are effective and reliable. The dissertation is divided into three chapters.In Chapter 1, we present a characteristic finite element approximation of quadratic optimal control problems governed by linear convection-dominated diffusion equations. In the first part, we consider the case that the control variable is under unilateral constraints, and the state and co-state variables are discretized by piecewise linear continuous functions and the control is approximated by piecewise constant functions. We derive some a priori error estimates for both the control and state approximations. It is proved that these approximations have convergence order (?)(h_U + h + k), where h_U and h are the spatial mesh-sizes for the control and state discretizations, respectively, and k is the time increment. The fundamental results of this part have been published in"Journal of Scientific computing"(see [32]).In the second part, we discuss another case where the control variable is under bilateral constraints and it is approximated by either piecewise constant functions or piecewise linear discontinuous functions. We also derive some a priori error estimates for both the control and state approximations. It is proved that these approximationshave convergence order (?)((?)+ h² + k) with m=0 or 1. The main results of this part have been submitted (see [33]). Numerical examples are given for both the two cases to check the convergence rate.In Chapter 2, we focus on deriving some a posteriori error estimates for both the control and state approximations which are of importance in developing adaptive characteristic finite element procedure for the control problems. By using the dual techniques we derive some corresponding L²-norm error estimates which consist of two parts. One partη₁ results from the approximation error of the control inequality (2.3.9), and the other partη_i,i=2,…,13 is contributed from the approximation error of the state and co-state equations. In particular,η₆ andη₁₁ correspond to the characteristic finite element approximations to the optimal control problems. It is clear that the states or control approximation errors alone cannot control the whole optimal control problems.In chapter 3, we mainly study another class of characteristic finite element (CFE) discretizations of optimal control problems governed by singularly perturbed, time-dependent convection-diffusion equations, where the non-divergence-free velocity fields are handled. In this chapter, the space discretization of the state variable is done by piecewise linear continuous functions, whereas the control variable is approximated by piecewise constant functions due to the limitation of the regularity. The goal of the present chapter is to derive a priori error estimates inε-weighted energy norm for the state and co-state approximations and L²-norm for the control approximation. Numerical experiments show that the new CFE method is also effective and reliable. Some results in this chapter have also been submitted.