| Optimal control problem of partial differential equations is a very lively and active mathematical field and also has been widely studied and applied in the last 30 years. This branch of mathematics covers various topics such as time optimal control,feedback control,analysis and control of flow equations,optimal shape design which are the models of applications in material design,in crystal growth,chemical reaction and others.The reader is referred to see examples of[59.69.72.75].It includes not only stationary and liner equations but also time-dependent and nonlinear ones.Especially, current research in the control of partial differential equations is driven by a multitude of applications in engineering and science that are modeled by coupled linear or nonlinear differential equations,for example,[4,22,36,37].Because of the complexity of problem in real applications,and what is also desirable and in fact necessary,to ensure that our mathematical theory can be widely and successfully applied to engineering and physical problems,it is a great challenge to form our optimal control models which are more reasonable and well-posed.On the other hand,with the development,of computer, we need to compute and to implement these control problems to guide our real performance in engineering and manufacture.Now the becoming second task is that associated optimal control problems need efficient numerical methods to deal with the resulting problems.It needs a fast development of numerical methods and the associated analysis must keep track to justify them and to prepare the basis for further research.Among these numerical methods,as well known the finite element method is a powerful tool.The finite element approximation of optimal control problems has been extensively studied in the literature.Furthermore,there have been extensive studies in convergence of the standard finite element approximation of optimal control problems,see,some examples in[5,6,31,52],although it is impossible to give even a very brief review here. For optimal control problems governed by linear state equations,a prior error estimates of the finite element approximation were established long ago;see,for example[27]. But it is more difficult to obtain such error estimates for nonlinear control problems. For some classes of nonlinear optimal control problems,a priori error estimates were established in[7,38.62].However,when the control is constrained it seems not easy to prove that the a priori error estimates for states or co-states are of optimal order. So in order to conquer that,superconvergence analysis is introduced as an important technique in finite element approximation of PDEs.For the approximation of PDEs, the recovery technique is popular in the superconvergence analysis.As optimal control problems to be concerned,due to the lower regularity of the constrained optimal control(normally only H1(Ω)∩W1,∞(Ω)),only a half order of convergence rate can be expected to gain by using the standard recovery technique,see[81].Very recently Meyer and RSsch in[68]showed that in fact one order can be gained via using a special projection,which was unique to the linear optimal control problem they studied.This is a quite interesting result considering the low regularity of the optimal control.Besides, [61]gives the related superconvergence result about the optimal control of Stokes problem.Prom the literature,we find that it is very useful to establish such a superconvergence property for optimal control problems,which is normally difficult to compute with higher accuracy,but it seems not to exist too many results for other kinds of optimal control problems.So,superconvergence analysis for optimal control problems is becoming an attractive and challenging topic in the field of numerical computing of control problems.There has been so extensive research on developing fast numerical algorithms for optimal control in the scientific literature that it simply impossible to give even a very brief review here.However there seems still some way to go before efficient solvers can be developed even for the constrained quadratic optimal control governed by an elliptic equation.The reason seems that there are so many computational bottlenecks in solving an optimal control problem numerically,and furthermore they are all closely related.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 the adaptive finite element methods are essential in using a posteriori error indicator to guide the mesh refinement procedure.Only the area where the error indicator is larger will be refined so that a higher density of nodes is distributed over the area where the solution is difficult to approximate.In this sense efficiency and reliability of adaptive finite element approximation rely very much on the error indicator used.There exist several concepts including residual and hierarchical type estimators,error estimators that are based on local averaging,the so-called goat oriented dual weighted approach,and functional type error majorants(cf.[3,9,10]).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. On the other hand,as far as the development of adaptive finite element schemes for optimal control problems for PDEs is concerned,much less work has been done.The goal oriented dual weighted approach has been applied to unconstrained problems in [11].Residual-type a posteriori error estimators for control constrained problems have been derived and analyzed in[29,41,43,63,64].Also suitable adaptive meshes can quite efficiently reduce the control approximation error.It has been recently found that suitable adaptive meshes can greatly reduce discretisation errors,see[11,12,55,64].If the computational meshes are not properly generated,then there may be large error around the singularities of the control,which cannot be removed later on.It is pointed out only recently that the error indicators derived for the approximation of the state equation,which have been widely used in adaptive finite element schemes for optimal control,are not necessarily efficient for computing the optimal control problems,and purposed built error estimators for the control problem are essential in such schemes,see[11]and[55]for example.Furthermore in a constrained control problem,the optimal control and the state usually have different regularity,and what is more,the locations of the singularities are very different. Usually the optimal control has only limited regularity(at most in H1(Ω)). This indicates that the current all-in-one mesh strategy may be inefficient.Adaptive multi-mesh;that is,separate adaptive meshes which are adjusted according to different error indicators,are often necessary.Particularly it seems to be important to use multi-set adaptive meshes in applying adaptive finite element method to computing optimal control,see[46,54].However it is much more complicated to implement adaptive computational schemes for evolutional control problems,see[45,56].This dissertation is some research work on the superconvergence analysis and residual type a posteriori error estimate for adaptive finite element method.The dissertation is divided into three chapters.Firstly,for a class of bilinear elliptic control problem,we establish finite element approximation and give the superconvergence analysis as well as the numerical tests verify the theoretical results.The related work about this part. are being prepared for publishing by "Journal of Computational Mathematics".Then. we give the a posteriori error estimate and implement the adaptive computing on the multi-mesh.Besides,the related work about this part have been submitted.Secondly, the superconvergence result is derived for Bénard optimal control problem governed by the thermally coupled incompressible Navier-Stokes equation which is crucial to many technological and scientific applications.Also,the related article about this part are being prepared for publishing by "Journal of Computational Mathematics".In the following part,we give the a posteriori error estimate for the Bénard problem.Thirdly, we establish an important model of optimal control problem governed by system of immiscible displacement,in porous media.Then,we show the summary of each chapter.To optimal control problems except the linear type or Stokes type there does not exist much work on the superconvergence analysis,so in the first part of chapter 1,we investigate a priori error estimates and superconvergence properties for a model optimal control problem of bilinear type,which includes some parameter estimation application. The state and co-state are discretized by piecewise linear functions and control is approximated by piecewiseconstant functions.We derive a prior error estimates and superconvergence analyses for both the control and the state approximations with a projection or interpolator different from[68].We also give the optimal L2-norm error estimates and the almost optimal L∞-norm estimates about the state and co-state.In the second part,we study adaptive finite element discretisation schemes for constrained optimal control problems.We propose to use adaptive multi-meshes in developing efficient algorithms for this problem.We derive equivalent a posteriori error estimators for both the state and the control approximation,which particularly suit an adaptive multi-mesh finite element scheme.The error estimators are then implemented and tested with promising numerical results.As far as the coupled nonlinear control problems are concerned,the control of viscous flow for the purpose of achieving some desired objective is crucial to many technological and scientific applications.The Boussinesq approximation of the Navier-Stokes system is frequently used as mathematical model for fluid flow in semiconductor melts.In many crystal growth technics,such as Czochralski growth and zone-melting technics,the behavior of the flow has considerable impact on the crystal quality.It is therefore quite natural to establish flow conditions that guarantee desired crystal properties. As control actions,they include distributed forcing,distributed heating,and others.For example,the control of vorticity has significant applications in science and engineering such as control of turbulence and control of crystal growth process.Optimal control problems for the thermally coupled incompressible Navier-Stokes equation by Neumann and Dirichlet boundary heat controls were considered in[34,37].Also, the time dependent problems were considered in the literature.In the first section of chapter 2,we consider the Bénard problem whose state is governed by the Boussinesq equations,which is crucial to many technological and scientific applications.Without the control constraint,the analysis of approximation about optimal control of the stationary Bénard problem was considered in[53],and it uses the gradient iterative method to solve the discretized equations.For the constrained control case,there seems to be little work on this problem.The first part of this chapter is concerned with the finite element approximation and superconvergence analysis of constrained optimal control problem of the stationary Bénard problem.As well known,the a posteriori error estimate for coupled nonlinear optimal control problems with control constrained is so difficult that we are currently not aware of the existence of such results devoted to this kind of control problems.However,the adaptive finite element of this kind problem is very important in real applications,so it is the strong motivation to spend our research on this point.In the second part of chapter 2,we give the a posteriori error estimate for the stationary Bénard problem using dual techniques which is the key to prove.Besides,when analyzing the errors in this part,it is clear that the states and control approximation errors alone cannot control the approximation errors of numerical coincident sets.Thus we also need to consider the measurement of the coincident set approximation errors which plays a crucial roles in deriving sharper a posteriori error estimators.The field of petroleum engineering is concerned with the search for ways to extract more oil and gas from the earth's subsurface.In a world in which an increase in production of tenths of a percentage may result into a growth in profit of millions of dollars,no stone is left unturned.A common technique in oil recovery,known as "water flooding",makes use of two types of wells:injection and production wells. The production wells are used to transport liquid and gas from the reservoir to the subsurface.The injection wells inject water into the oil reservoir with the aim to push the oil towards the production wells and keep up the pressure difference.The oilwater front progresses toward the production wells until water breaks through into the production stream.An increasing amount of water is produced,while the oil production rate diminishes,until at some time the recovery is no longer profitable and production is brought to an end.Due to the strongly heterogeneous nature of oil reservoirs,the oil-water front does not travel uniformly towards the production wells,but is usually irregularly shaped.As a result,large amounts of oil may be still trapped within the reservoir as water breakthrough occurs and production is brought to an end.Using water flooding,up to about 35 percent of the oil can be recovered economically.The introduction of so-called "Smart" or "Intelligent" wells is one of the most promising developments in this field over the recent years.These types of wells allow for advanced downhole measurement and control devices,which expand the possibilities to manipulate and control fluid flow paths through the oil reservoir.The ability to manipulate the progression of the oil-water front provides the possibility to search for a control strategy that will result in maximization of oil recovery.A straightforward approach to find such a control strategy is to use a dynamic optimization technique to increase recovery by delaying water breakthrough and increasing sweep,based on a predictive reservoir model.Obviously,this problem can be described as an optimal control problem of PDEs.Meantime,the most realistic way to control the flow paths through the reservoir is to manipulate the the quantity of water injected by the control valve settings.Based on this point,with the aim to maximize production of oil from petroleum reservoirs,we construct the optimal control problem.Reservoir simulators use conservation of mass and momentum equations to describe the flow of oil,water or gas through the reservoir rock.Although oil consist of a large number of chemical components with varying properties,in many reservoir modeling cases the Black Oil Model is adopted for simplicity reasons.This model distinguishes between three phases:water,oil and gas.For further simplification,in the oil reservoirs models used within this work no gas is assumed present,hence reducing the number of phases to two.So chapter 3 is devoted to the optimal control problem for the two-phase incompressible flow in porous media.Firstly,the formulation of our model is established which is an optimal control problem governed by elliptic coupled parabolic system.Then,the existence of solutions is stated and proved.The co-state equations are deduced and we also present the optimality condition.Furthermore,the state and co-state systems are discretized by finite element method.After that,the proof of existence about the solutions of adjoint equations is shown.
|
PDF Full Text Request