Domain Decomposition Methods And Characteristics Methods For Parabolic Problems

Mathematical physics and engineering problems can be turned into the problems of solving large partial differential equations. such as the exploration and development of oil and gas reservior, the large structure of engineering, the design of large spacecraft, aerodynamics, reactor etc.. The domains they are defined on are always large area with high dimension and irregular geometry, which cause many difficulties to computations when seeking their approximate solutions. Domain decomposition methods were new research direction which grew up in 1980s'. In short, domain decomposition methods divide the whole domainΩinto several sub-domains:(?), and the shapes of these sub-domainsΩ_i may as well as regular. And then the solutions of the original problems can be translated into solving the questions on the sub-domains respectively. Because they have many advantages, such as they can divide large-scale problems into several small ones, complex boundary problems into several simple boundary ones and successive problems into parallel ones, etc., which other methods don't have, they rapidly become the hot field of computational mathematics.Domain decomposition methods may be partitioned into two families: overlapping domain decomposition methods and nonoverlapping domain decomposition methods. The selection of sub-domains may be based on considerations of available computing region shape and physical background of the problems. The latter is, in particular, applicable to complex systems which consist of possibly different governing equations in different physical sub-domains. Overlapping domain decomposition methods become harder to implement in such a setting, while the nonoverlapping domain decomposition methods may be more directly applicable. In addition, the theoretical analysis of the nonoverlapping domain decomposition methods are more difficult.In this dissertation, our researches in domain decomposition methods are focussed on overlapping domain decomposition methods. The initial idea of overlapping domaindecomposition methods came from the classical Schwarz alternating algorithm. In recent years the theoretical researches and applications on domain decomposition methods based on Schwarz alternating algorithm have been developed adequately so that these methods become very powerful and efficient iterative methods. A systematic theory has been developed for elliptic finite element problems in the past few years [6, 14, 16]. Lions [40, 41] presented a kind of Schwarz alternating algorithm in two sub-domain case for heat equations and gave a convergence result but did not give any error estimate. X. C. Cai [13, 15, 16, 17] constructed a kind of additive Schwarz algorithms and multiplicative Schwarz algorithms and prove that the convergence rate is smaller than one for parabolic equations, where the author did not consider the dependence of the convergence and the discretization parameters. H. Rui and D. Yang [50, 51] considered how the convergence and the error estimate depend on the diameter of sub-domains,the spacial mesh-size, the time step increment and the number of iterations at each time level.In this dissertation, we also do some researches on the method of characteristics for convection-diffusion equations. Convection-diffusion equations describe miscible displacement flow processes in petroleum reservoir simulation, subsurface contaminant transport and remediation, disposal of nuclear waste in underground repositories, and many other applications. Their solutions typically have moving steep fronts that need to be resolved accurately. Standard finite difference or Galerkin finite element methodstend to yield numerical solutions with severe nonphysical oscillations. The classical upwind finite difference method greatly reduces these oscillations but introduces excessivenumerical dispersion. Many extensive schemes have been carried out to overcome these difficulties and allow accurate numerical solutions with reasonable computational effort, such as the methods of characteristics [24, 25, 26, 35, 48, 49].In 1982, J. Douglas, Jr. first developed the method of characteristics [26]. It is effective to approximate the solutions of convection-diffusion equations by using the method of characteristics. Since then, J. Douglas, Jr., R. E. Ewing, T. F. Russell, M. F. Wheeler, T. Arbogast and Y. Yuan have completed a series of fundamental researches in this field [24, 25, 35, 48, 49, 64, 65]. From the numerical experiments, we could find that characteristics methods could increase time step and avoid numerical oscillation, as opposed to general methods. In [52], Rui and Tabata first introduced a second order characteristic finite element method. The numerical scheme is of second order accuracy in time increment, symmetric and unconditionally stable.Under the aborative guidance of my supervisor Professor Hongxing Rui, the authorhas finished this dissertation consisting of work on domain decomposition methods and characteristics methods. Firstly, based on additive Schwarz algorithms proposed by X. C. Cai, we present some additive Schwarz methods and analyze the convergenceof each procedure. And we study the dependence of the convergence rate on the spacial mesh size, the time increment, the sub-domains overlapping degree and the number of iterations at each time level. Both theoretical analysis and numerical experiments show that all these algorithms are high parallelism. Secondly, based on multiplicative Schwarz methods proposed by X. C. Cai and combined with a second order characteristic finite element scheme, a new kind of domain decomposition meth-ods are constructed for convection-diffusion equations. The analysis of stability and convergence of these algorithms is given. Finally, we apply a characteristics-mixed finite element method, which was proposed by T. Arbogast and M. F. Wheeler, to solve unsteady-state convection-diffusion equations. We prove the convergence of the scheme. The error estimate holds uniformly with respect to the vanishing diffusion coefficient.The dissertation is divided into three chapters.In Chapter 1, we firstly consider the solution of linear systems of algebraic equationsthat arise from parabolic finite element problems. We present additive Schwarz domain decomposition methods for parabolic problems and also consider the dependenceof convergence rates of these algorithms on parameters of time step and space-mesh.The resulting linear systems of equations are symmetric positive definite and solved by the conjugate gradient method. Both theoretical analysis and numerical experimentsshow that all these algorithms are high parallelism. Then, we introduce a kind of modified additive Schwarz methods also with the conjugate gradient method, which are more suitable for parallel computers than anterior algorithms. We analyze the convergence of modified additive Schwarz algorithms. Numerical experiments also confirm the efficiency and superiority of the algorithms. The part of the results in this chapter have been published in the Applied Mathematics and Computation (see [46]).In Chapter 2, we mainly study the domain decomposition methods for convection- diffusion equations. We know that the solutions of convection-diffusion equations typicallyhave moving steep fronts. Standard finite difference or Galerkin finite element methods for parabolic problems tend to yield numerical solutions with severe non-physicaloscillations. Based on multiplicative Schwarz methods proposed by X. C. Cai and combined with a second order characteristic finite element scheme, a new kind of domain decomposition methods are constructed for convection-diffusion equations. The stability and convergence analysis of these algorithms is given. We analyze the dependence of convergence rates of these algorithms on parameters of time step and space-mesh.In Chapter 3, we study a characteristics-mixed finite element method for unsteady-stateconvection-diffusion problems. The equation, which we consider in this chapter, is convection-dominated and the equation is nearly hyperbolic in nature. The concentrationoften develops sharp fronts that are nearly shocks. Because of the hyperbolic nature of convective transport, characteristic analysis is natural to aid in the solution of convection-diffusion equations. Effective discretization schemes recognize to some extent the hyperbolic nature of the equation. Many such schemes have been developed, such as the explicit method of characteristics, the streamline diffusion method [38], the modified method of characteristics-Galerkin finite element procedure (MMOC-Glerkin) [24, 26, 31], and the Eulerian-Lagrangian localized adjoint method (ELLAM) [18, 19, 54, 56, 57]. In this chapter, we develop a characteristics-mixed finite element method to solve unsteady-state convection-diffusion equations with all combinations of general inflow and outflow boundary conditions. Our test functions are piecewise constant in space, and in time they approximately follow the characteristics of the convective (i.e., hyperbolic) part of the equation. Thus the scheme uses an approximate characteristics to handle advection in time. And we use a lowest-order Raviart-Thomas-Nedelec mixed finite element spatial approximation of the equation. Boundary conditions are incorporatedin a natural and mass conservative fashion. The scheme is completely locally conservative; in fact, on the discrete level, fluid is transported along the approximate characteristics. The error estimate holds uniformly with respect to the vanishing diffusioncoefficient.