Shandong University Doctoral Dissertation Numerical Methods And Analysis For System Of Parabolic Equations

A large number of physical problems of significant interest are modelled by the systems of parabolic equations. Examples include the simulation of multiphase and multi-component flows in porous media coming from petroleum and environmental engineering, of certain semiconductor devices, of nuclear waste-disposal contamination problems, of the dynamics of competing biological species, and of various atmospheric and fluid flows. The investigation of the systems of the parabolic equations has been the object of an increasing interest of a number of specialists, engineers as well as mathematicians. There are a great deal of literatures on analytical results about the systems, e.g., existence and uniqueness of the solution~[1-10] as well as on numerical results, e.g., convergence and error estimate of the finite element methods(FEMs) and the finite difference methods(FDMs). The theory and the computer simulations show that the systems of parabolic equations are interesting from a mathematical point of view as well as from a biological, chemical, physical point of view.Kalashnikov A.S.,Zhou Yulin, Shen Longjun, Yuan Guangwei, Wu Zhuoqun, et al. paid special attention particularly to pure theory analysis on the properties of the solu-tion to the system of parabolic equations. While J. Douglas Jr., Dupont, R.E. Ewing, T.F.Russell, M.F.Wheeler and Yuan Yirang, et al.~[11-20], completed a series of fundmen-tal research on numerical analysis for approximate solution to the system of parabolic equations. They put forward many famous numerical methods, such as the characteristic finite difference method, the characteristic finite element and the characteristic mixed finite element method etc.In the period of 1990s, the numerical methods and theories developed more adequately.In this dissertation, we consider system of parabolic equations and convection-diffusion problems,by comprehensively employing finite element method (FEM) and finite volume element method(FEVM) the two discretizing techniques, and put forward corresponding schemes. We give rigorous convergence analysis and experiments which verify the theoretical results and indicate the efficiency and validity of these schemes.Some new workswe do in thesis are as foolows:(l)We put forward alternating direction multistep method(ADIMSM) and finite volume element predictor-corrector method(FVEPCM) for linear and nonlinear systems of parabolic equations, respectively; derived optimal L2-norm error estimates and gave experiments for both schemes; meanwhile derived energy norm error estimate for the latter scheme.(2) Based on triangulation and arbitrary quadrilateral mesh partition and their dual partition, we put forward upwind finite volume element schemes for 2-D diffusion problems with nonlinear convection terms. The schemes possess characters of mass conservation and small computational cost. We derived optimal L2-norm and H1-norm error estimates, respectively,and gave experiments which support theory analysis for both schemes. We pointed out that the latter scheme satisfies discrete maximum principle.(3) Based on straight triangular prism partition and cuboid mesh partition and their dual partitions, we put forward upwind finite volume element schemes which combined with lumped masses operator for 3-D nonlinear convection diffusion problems. The schemes possess characters of mass conservation and small computational cost. We derived optimal L2-norm and i^-norm error estimates, respectively,and gave experiments which show the schemes are feasible . We pointed out that the latter scheme satisfies discrete maximum principle.This paper is divided into five chapters.In the first three chapters we consider linear and nonlinear systems of parabolic equations, and put forward ADIMSM, moving finite element method (moving FEM) and FVEPCM. In the last two chapters we study 2-D and 3-D diffusion problems with nonlinear convection terms, and construt upwind finite volume element schemes based on four kinds spaitial partition.In the first chapter, we consider a class of linear parabolic system, put forward and analyze ADIMSM, do some numerical experiments which show that the method is high efficient. Since their introduction in a finite difference context, alternating direction implicit (ADI) methods have proved to be valuable techniques for the efficient solution of parabolic and hyperbolic initial-boundary value problems in several space variabW22~35lln 1971, Douglas and Dupont formulated Galerkin alternating-direction procedures for nonlinear parabolic equations and linear second-order hyperbolic problems posed on a rectangular region with a uniform grid t23' and derived optimal Hq -error estimates for each method. These alternating-direction procedures are of interest because by their use one can solve large multidimensional problems as a series of smaller one-dimensional problems, and the matrices which must be inverted at each time step of the solution process are independent of time and require only one decomposition. The storage requirements for these matrices are associated with one-dimensional problems rather than the full multidimensional problem, so the storage requirments can be quite low. The Galerkin alternating-direction methods are particularly attractive for solving large three-dimensional nonlinear problems. Dendyt31' derived optimal L2-error estimates for several ADIschemes proposed by Douglas and Dupont for parbolic problems, and also for schemes for nonlinear second-order hyperbolic problems which are extensions of methods of Douglas and Dupont. Fairweather'34] presented an ADI Galerkin method for a linear fourth-order parabolic problem, while Dryja^3^ discussed such methods for a linear system of first-order hyperbolic equations and a system of parabolic- hyoerbolic equations.ADI Galerkin methods were extended to rectangular polygons domain by Dendy and Fairweathert32!, and El-Zorkany and Balasubramanianf35l Generalizations of ADI Galerkin methods to parabolic problems on nonrectangular, curved regions domain were presented by Hayes'24'25!. J. H. Bramble, R. E. Ewing, and Li Gang'36! combined ADI with multi-step method and presented ADI multistep method for a scalar parabolic problem and derived optimal L2-error estimates.The purpose of this chapter is to formulate and analyze a new ADI Galerkin multistep method for system of linear parabolic equations initial-boundary value problems.A brief outline of this chapter is as follows. §1.2 consists of §1.2.1 and §1.2.2. In §1.2.1, we introduce some notation ,assumptions and preliminaries and present ADI multistep method in §1.2.2. In §1.3, optimal L2-error estimate is derived. Numerical experiment which verifies that the method is high effective is given in §1.4.In chapter 2, we consider a class of full nonlinear parabolic system, put forward and analyze MFEM. The MFEMs have important applications for a variety of physical and engineering areas such as solid and fluid dynamics, combustion, heat transfer, material science, etc. The physical phenomena in these areas develop dynamically singular or nearly singular solutions in fairly localized regions,such as waves, boundary layers, detonation waves, etc. The numerical investigation of these physical problems may require extremely fine meshes over a small portion of the physical domain to resolve the large solution variations.The use of moving FEMs can greatly improve the accuracy of the numerical approximations and also decrease the computational cost. In the past three decades, there has been important progress in developing finite element methods with moving mesh for PDEs. R.Bonnerot and P.Jamet^ proposed a space-time finite element method which is a veryuseful finite element method with moving mesh in 1974, firstly. In 1981, K.Miller and R.N.Millert39>40l proposed and analyzed moving finite elements for one-demension Burgers' equation with a large Reynolds number. Liang^41>42>43^ put forward moving FEM for linear parabolic equations and derived optimal L2-norm convergence rate. Yuanf44'45*46' proposed moving FEM for two-phase incompressible displacement in porous media. Moreover, there are moving FEMs for other type of equations, such as hyperbolic equation^47'48',etc..A brief outline of this chapter is as follows. §2.1 is introduction, in which the moving FEMs is reviewed, The considered problem and corresponding suppositions are given. Based on the discretizing technique of moving FEM, §2.2 gives a kind of high efficiency moving FEM. In §2.3, optimal L2-norm and energy norm error estimates are derived.In the following three chapters, we will pay more attentions to finite volume element methods(FVEMs). Firstly, let us briefly describe the FVEM. The FEVM is a discretization technique for paratial differential equations, especially for those, that arise from physical conservation laws including mass, momentum and energy. This method has been introduced and analyzed by R. Li and his collaborators since 1980s,see [88] for detail. The FVEM uses a volume integral formulation of the original problem and a finite partitioning set of covolumes to discritize the equations.The approximate solution is chosen out of a finite element spaces [51>52>881. The FVEM is widely used in computational fluid mechanics and heat transfer problems t5°-52>87].lt possesses the important and crucial property of inheriting the physical conservation laws of the original problem locally. Thus it can be expected to capture shocks, to produce simple stenciles, or to study other physical phenomena more effectively.In chapter 3, we consider a class of full nonlinear parabolic system, put forward and analyze the finite volume element predictor-corrector method(FVEPCM). The FVEMs which provide effective discrete schemes for partial differential equations(PDES),especially in computational fluid dynamics,are widely used for their conservation property of the original problem. In recent years,FVEMs have been used to solve scalar PDES(49H71l,but never to solve the parabolic system of equations.In [72] Yang only put forward FVEM for the 1-d parabolic system of equations,but didn't give convergence analysis.In [37] Douglas and Dupont put forward a family of predictor-corrector-Galerkin procedures for the parabolic equations and obtained Hl error estimates.Wheeler in [73] derived optimal L2 error estimates for continuous time and several discrete time Galerkin approximations of some second order nonlinear parabolic boundary value problems.In this chapter, the FVEPCM which is used to solve a class of nonlinear parabolic system of equations is put forward. In §3.2, we put forward FVEPCM. Several usefulauxiliary lemmas are given in §3.3 and optimal L2 error estimate is derived in §3.4. At last in this chapter we present a numerical experiment which shows that numerical results are consistent with the theory analysis.In chapter 4 and chapter 5, we will turn to consinder convection-diffusion problem. The convection-dominated diffusion problem has strong hyperbolic characteristics, and therefor the numerical method is very difficult in mathematics and mechanics, when the central difference method , though it has second-order accuracy , is used to solve the convection-dominated diffusion problem, it produces numerical diffusion and oscillation, making numerical simulation failure. The case usually occurs when the finite element methods(FEM) and FVEM are used for solve the convection-dominated diffusion problem.For the two-phase plane incompressible displacement problem which is assumed to be Q- periodic, Jim Douglas,Jr.,and T.F.Russell have published some articles on the characteristic finite difference method and FEM to solve the convection-dominated diffusion problems and to overcome oscillation and faults likely to occer in the traditional method^74'. Tabata and his collaborators have been studying upwind schemes based triangulation for convection-diffusion problem since 1977t75~79lYuan Yirang ,starting from the practical exploration and development of oil-gas resources , put forward the upwind finite difference fractional steps methods for the two-phase three-dimensional compressible displacement problem'86'.Most of the papers known concernes on the FEVM for one- and two-dimensional linear partial differential equations^50"52'63"71]. In recent years, M.Feistauer, J.W.Hu, etcJ81~85^, by introducing lumping operator, constructed Finite Volume-Finite Element Method for Nonlinear Convection-Diffusion Problems. On the other hand, because FEM costs great expense to solve the three-dimensional problems, we usually use finite difference method(FDM) to approximate the problems^. These works inspire us to look into the subject how to use upwind FVEM to solve three-dimensional nonlinear convection-dominated diffusion problems. In the two chapters , we consider two cases: 2-dimensional and 3-dimensional. we put forward two kinds of upwind FVEMs for 2-D nonlinear convection-dominated diffusion problems based on triangulation and quadrilateral net partition and their dual partitions of fi, respectively, and based on straight triangular prism partition and cuboid mesh partition and their dual partitions of ￡1 for 3-D case, respectively. Such as calculus of variations, commutating operator and prior estimates, are adopted. Optimal order error estimate in L2 for the first scheme of the two cases and Discrete maximum principle and optimal order error estimates in Hl for the second scheme of the two cases are derived to determine the error in approximate solution .For simplifyingcomputing costs, in the second method for 3-D case we introduce lumping operator. Numerical experiments show that the methods are effective for avoiding numerical diffusion and nonphysical oscillations and reducing work time.The outlines of the two chapters are organized as follows.In chapter 4, we consider 2-D convection-diffusion problem.§4.1 is introduction, we breifly review the FVEM and the convection-diffusion problem. In §4.2 we put forward and analysis the upwind finite volume element method for problem (4.1.1) based on arbitrary triangulation and its dual partitions of ft. In §4.2.1,we introduce notation construct triangulation Th of fi and give the dual partition. Some Auxiliary Lemmas and the optimal order error estimation in L2-norm of the scheme are shown In §4.2.2 and §4.2.3 respectively. In §4.2.4, Numerical experiment shows that the method is effective for avoiding numerical diffusion and nonphysical oscillations. In §4.3 we put forward analysis the upwind finite volume element method for problem (4.1.1) based on arbitrary quadrilateral mesh partition and its dual partitions of fi. In §4.3.1,we introduce notation ,construct arbitrary quadrilateral mesh partition T/, of fi and give the dual partition. Based on the case of rectangular meshes, the discrete maximum principle and the optimal order error estimation in K^norm of the scheme are shown In §4.3.2 and §4.3.3 respectively. In §4.3.4, numerical experiment shows that the method is effective for avoiding numerical diffusion and nonphysical oscillations.In chapter 5, we consider 3-D convection-diffusion problem. The corresponding contents of FVEM and the convection-diffusion problem are reviewed in §5.1 Introduction. In §5.2 we put forward and analyze the upwind finite volume element method for problem (5.1.1) based on straight triangular prism and its dual partitions of f2. In §5.2.1,we introduce notation ,construct straight triangular prism mesh partition T/, of fi and give the dual partition. Some Auxiliary Lemmas and the optimal order error estimation in L2-norm of the scheme are shown In §5.2.2 and §5.2.3 respectively. In §5.2.4, Numerical experiment shows that the method is effective for avoiding numerical diffusion and non-physical oscillations. In §5.3 we put forward analysis the upwind finite volume element method for problem (5.1.1) based on cuboid mesh partition and its dual partitions of H. In §5.3.1,we introduce notation ,construct cuboid mesh partition 7X of Q. and give the dual partition. The discrete maximum principle and the optimal order error estimation in i^-nonn of the scheme are shown In §5.3.2 and §5.3.3 respectively. In §5.3.4, the results of numerical experiment are given.