Continuous Space-time Finite Element Method And Numerical Simulation For Evolution Type Equations

The space-time continuous finite element method is a kind of high accuracy numerical method,which uniformly handles the spatial variables and temporal variable.Namely,this method not only uses finite element to discretize spatial variables but also uses it to discretize temporal variable.Thus,compared with the classical finite element methods,it is more easy to obtain high precision with respect to time and its theoretical analysis will not change with the discretization ways of temporal variables,namely it holds uniformly for approximate polynomials of any degrees.What's more,this method is particularly suitable for wave problems as its discrete form with important energy conservation.The space-time continuous finite element method can be divided into the following two cases.The first one is that each time level corresponds to the same spatial partition;the second one is that each time level can correspond to different space mesh structures.For case 1,since each time step has the same space partition,after introducing space-time projection operators in entire time interval we can easily obtain error estimates of space-time finite element solution in various norms.For case 2,since each space-time slab allows variable space-time mesh structures,it is particularly appropriate for adaptive computations on unstructured meshes.In addition,for case 2 we respectively introduce the Lagrange interpolation polynomials fixed by Legendre points and Lobatto points as well as the corresponding Gauss integration rules,and we take full advantages of the basic properties of Lagrange interpolation polynomials and the high precision characteristic of Gauss integration in theoretical analysis,which make the theoretical analyses more natural and easy to understand as well as capture the essence of time stepping finite element method at a deeper level.Additionally,whether the case 1 or case 2,its continuous space-time scheme is often unconditional stable and its theoretical analysis is usually not restricted by space-time mesh,that is,it does't need time steps and space grid parameters to meet certain conditions.This article mainly from the two aspects of theoretical analysis and numerical simu-lation to study the space-time continuous finite element method under the case 1 and case 2 for time-dependent partial differential equations.The Chapter 1 is the introduction,which mainly expounds the research status of the space-time continuous finite element method as well as the research contents of this article and article structure.In addition,it gives some preliminaries which are necessary for theoretical analyses of this thesis.In Chapter 2 and Chapter 3,we use space-time continuous finite element method to study the Sobolev equations and viscoelastic wave equations under case 1,respectively.We first construct the space-time continuous scheme for the original problem and prove the existence,uniqueness and stability to the space-time continuous solution,and then through the introduction of space-time projection operators and the use of relatively simple theoretical analysis gives the L2 and H1 norm estimates at the time nodes as well as global L2(L2)and L2(H1)norm estimates.And we also need to point out that the estimates given here don't need any restrictive conditions between the space and time grid size.Finally,we give a two-dimensional numerical example on unstructured meshes to confirm the effectiveness and feasibility of the scheme.Furthermore,the numerical example also show that compared with the traditional finite element method where the time variable is discretized by Euler or Crank-Nicolson(CN)scheme,the space-time continuous finite element method is more easy to obtain the high precision concerning space and time.In Chapter 4,we study the wave equation by the space-time continuous finite element method under case 1.In this Chapter,a new space-time continuous finite element method which is used to solve the wave equation is proposed.We first get the scheme which is equivalent to the original space-time continuous scheme by introdueing the Legendre polynomial and its corresponding Gauss quadrature rule,then we analyze the existence and uniqueness of the approximate solution based on this scheme.Moreover,we give the L2 and H1 norm estimates to the the approximate solutio1 at the time nodes via introducing the space-time projection operators.At last,two numerical examples are provided to validate the effectiveness and feasibility of the algorithm proposed here.Comparing with the existing methods,the analytical method given here is more concise and understandable and it is easy to be extended to other wave problems.At the same time we need to point out that in the process of solving the viscoelastic wave equation and wave equation,we first obtain a coupled system which is equivalent with the original problem through introducing the auxiliary function v?vt,then we construct the space-time continuous scheme based on this coupled system,through solving this scheme we can obtain the high accuracy regarding u and v simultaneously.In Chapter 5 and Chapter 6,we respectively apply the space-time continuous finite element method with mesh modification to study the Sobolev equations and viscoelas-tic equations under case 2.We first construct the space-time continuous schemes with grid change to the original problems,which are regarded as a kind of extension to the schemes proposed in Chapter 2 and Chapter 3.Then we give the well posedness analysis of the numerical solutions via introducing the Lagrange interpolation polynomials fixed by the Legendre points and the Gauss-Legendre quadrature rule;giving the L?(L2)and L?(H1)error estimates of the numerical solutions by introducing the Lagrange interpo-lation polynomials determined by the Lobatto points as well as corresponding integration rule.In addition,in Chapter 6 we also prove that if the grid in each time level satisfies some reasonable assumptions,then the jump terms in the convergence results can be elim-inated such that we can obtain optimal order estimate inL?(L2)norm with respect to time and space.In Chapter 7,the convection-dominated Sobolev equation with variable coefficient is studied under case 2.We prove the existence and uniqueness to the numeri-cal solution and give the optimal order estimate in L?(L2)norm without space-time mesh restriction.Finally,we give the numerical simulations for the original problem in the case of the space-time continuous finite element scheme and time discontinuous space-time finite element scheme,respectively.The numerical experiments confirm the correctness of the analysis and present that the space-time continuous finite element method is more effective than the time discontinuous space-time finite element method in practical com-putations.