The Numerical Discrete Methods For Viscous Incompressible Flow Problems And Maxwell's Equations

The study of numerical discrete methods for viscous incompressible flow problems is the hot topic in computational mathematics.Navier-Stokes equations are the basic of viscous incompressible fluid mechanics.Stokes equations are the stationary form and linearization of Navier-Stokes equations.Therefore,it is typical and universal signif-icances to study the numerical methods for Stokes equations.The Brinkman equations are used to describe the flow in complex porous media with highly varying permeability coefficient.It is also important to develop the stable numerical discrete schemes for the Brinkman equations.The mixed finite element method is an efficient numerical method for solving vis-cous incompressible flow problems.However,on the one hand,the conventional mixed finite element method requires the pair of finite element spaces to satisfy the inf-sup sta-bility condition.This condition limited the application of simple lower-order polynomial space pairs.On the other hand,the shape of mesh decomposition is generally triangular or quadrilateral?9?=2)elements.This makes it difficult to satisfy the stability condition and solve the problems in complex boundary region.In addition,the numerical solution of viscous incompressible flow strictly satisfies the incompressible conditions,which is of great significance to the stability and convergence of the solution.However,it is diffi-cult to construct the divergence-free finite element scheme by the traditional mixed finite element methods.In recent years,the research of numerical discrete methods for viscous incom-pressible flows has turned to non-standard finite element methods,such as discontinu-ous Galerkin finite element method,hybridizable discontinuous Galerkin finite element method and weak Galerkin finite element method,etc.The advantages of these meth-ods are flexible mesh generation,easy to satisfy stability conditions and construct finite element schemes satisfying divergence-free.The fist work in our dissertation is the weak Galerkin finite element methods for two kinds of viscous incompressible flow problems–the Stokes equations and the Brinkman equations.We develop a globally divergence-free method for these two problems respec-tively,prove the stability of the schemes and obtain uniform error estimates with respect to the Reynolds number.Finally,the numerical experiments confirm the theoretical re-sults.The second work in our dissertation focuses on numerical analysis of Maxwell's equations.The Maxwell's equations are the basic equations in electromagnetic,and nu-merical methods of Maxwell's equations have always been a hot topic in computational electromagnetic,and are applied in electronic industry.Finite-Difference Time-Domain Method?FDTD?is one of the most popular numerical methods.However,the FDTD method is conditionally stable,so the time step and space step need to satisfy the Courant-Friedrichs-Lewy?CFL?stability condition.When electrically small problems are solved,space step must be small enough,according to the CFL condition,then time step should be small correspondingly,which increases the computational complexity,prolonging the computational time,and sometimes impossible.Therefore,many unconditionally stable difference methods are proposed,such as the ADI-FDTD method?Alternating Direction Implicit Finite-Difference Time-Domain Method?,Split Finite-Difference Time-Domain Method?Split Finite-Difference Time-Domain Method?,Symplectic methods,etc.On the other hand,energy conservation is an important property of electromagnetic field-s.For example,the electromagnetic energy of classic Maxwell's equations in lossless medium without source is preserved.Therefore,it is of practical significance to propose a stable and energy-conserving FDTD method for Maxwell model in lossless medium.We study the finite difference time domain method for two-dimensional Maxwell's equations in our dissertation.One is that we propose and analyze two unconditional stable schemes for the Maxwell's equations.The other is that we analyze the energy of the ADI-FDTD method with fourth order accuracy in time for the two-dimensional Maxwell equations in the lossless medium without sources and charges.The following is the main structure of our dissertation.In Chapter 1,we introduce the backgrounds and the existing numerical methods of the viscous incompressible flow problems and the Maxwell's equations.In Chapter 2,we introduce some definitions and notations of Sobolev space,some important formulas and the definition of weak gradient operator and weak divergence operator.In Chapter 3,we present a modified weak Galerkin method for the Stokes equations.The modified method uses the P₆)??/P₆-1)???6??1)discontinuous finite element com-bination for velocity and pressure in the interior of element,and the numerical traces₍??7?which are defined in the interface of the elements belong to the space⁰??_??.The stability,priori error estimates and²error estimates for velocity are proved in this dis-sertation.In addition,we prove that the modified method also yields globally divergence-free velocity approximations and has uniform error estimates with respect to the Reynolds number.Finally,numerical results show that the modified method has less degree of free-dom for the resultant linear system than the original method,and is more efficiency than the original method.In Chapter 4,we present and analyze a new weak Galerkin?WG?finite element method for the Brinkman equations.The variational form we considered is based on gradient–gradient operators,which is different from the others.The stability,priori error estimates and²error estimates for velocity are proved in this dissertation.In addition,we prove that the new method yields globally divergence-free velocity approximations.The convergence rates are independent of the Reynolds number.Finally,the numerical experiments confirm our theoretical results.In Chapter 5,firstly,we propose two schemes TS-FDTDI and TS-FDTDII for the Maxwell's equations.This two schemes are based on the splitting finite-difference time-domain methods and interpolation.The stability and error estimates are proved.Second-ly,we analyze the energy of the ADI-FDTD method with fourth order accuracy in time for the two-dimensional Maxwell's equations in the lossless medium without sources and charges,and obtain the numerical energy identity.In comparison with the energy in theo-ry,the numerical one has two perturbation terms and can be used in computation in order to keep the approximate energy conservation.Finally,numerical experiment is provided and numerical results confirm the theoretical results.