Mixed Space-time Discontinuous Finite Element Method For Evolution Type Equations

Unifying the space and time variables, time discontinuous space-time finite element method overcomes the low order accuracy in traditional finite element method caused by the difference discretization in time. This method has high-order accuracy in space and time directions, good dissipation on unstructured mesh, unconditional stability. Thus it becomes an efficiency method for the problems dependent on time.In recent years, hybrid applications of two or more numerical methods are more popular, and many good results have been derived. A high precision, effective numerical procedure can be obtained form hybrid application of two numerical methods. And it takes full advantages of both methods in numerical calculations. In this thesis, we mainly study from the aspects of the theoretical analysis and numerical computing of the high precision mixed numerical method, combining the time discontinuous space-time finite element method with other finite element methods, for some kinds of evolution type equations (including pseudo parabolic differential-integral equation, convection-diffusion equation, telegraph equation, pseudo hyperbolic equation and nonlinear Sobolev equation).A time discontinuous space-time finite element method is proposed and analyzed by Karakashian and Makridakis (1998) for the nonlinear Schordinger equation. The existence of the approximate solution and the optimal-order error estimate in L∞(L2)-norm are proved by using the technique of combining finite difference and finite element methods. In this technique, the feature of Lagrange interpolation polynomials on Radau points in time discretization interval In is applied to avoiding the restrictions on the space-time mesh. This technique is more appropriate to analyze the model problems which dose not have "strong" stability properties, and is valid under weak restrictions on space-time mesh. The stability and convergence theorems of the mixed numerical schemes proposed in this thesis are provided mainly by using this technique. In Chapter1, the features of the time discontinuous space-time finite element method are summarized, and the development and applied foreground of this method are introduced briefly. The preliminaries associated with this thesis are presented in Chapter2. In Chapter3, the time discontinuous space-time finite element method for a gen-eral pseudo parabolic differential-integral equation is studied. By introducing L2(Ω)-projection operator, the optimal priori error estimate in L∞(H1)-norm is derived. The regularity of the solution required in the error estimate obtained by using L2(Ω)-projection operator is lower than that required in error estimate obtained by using H1(Ω)-projection operator.In Chapter4, a mixed high precision numerical method, combining the time discon-tinuous space-time finite element method with H1-Galerkin method, is introduced and analyzed for a kind of convection diffusion problem. The optimal priori error estimate in L∞(H1)-norm is derived. Finally, two class of convection-diffusion problems are simulated by approximating the space integrals by composite two-point Gauss quadrature rule and using a cubic spline trial space and a piecewise linear test space in spatial direction to confirm the validity of the proposed method. One practical advantage of this method over the orthogonal cubic spline collocation method is that there only a half the number of unknowns in spatial direction under the same partition, and, therefore, it reduce the size of the matrix and hence, the computational cost. This method takes full advantages of the two method in numerical calculations, and achieves both high calculation efficiency and precision.In Chapter5, we formulate "two fields" time discontinuous space-time finite element method for telegraph equation based on the natural framework of second-order hyperbolic equations. In this scheme, displacement υ and velocity ut are approximated simultane-ously, and the continuities of displacement and velocity in time within each space-time partition unit are enforced through the strain energy inner product. The stability and optimal priori error estimate of displacement in L∞(H1)-norm and of velocity in L∞(L2)-norm are proved by using the technique of taking full advantages of both finite difference and finite element methods. Furthermore, numerical experiments are given to confirm the theoretical results.In Chapter6and Chapter7, a splitting mixed space-time discontinuous finite element procedure for a kind of parabolic problem and pseudo hyperbolic problem is formulated by combining the time discontinuous space-time finite element method with the split-ting positive definite mixed finite element method and H1-mixed finite element method, respectively. For parabolic problem, the optimal priori error estimates of displacement u in L∞(L2)-norm and of stress σ=-A▽u in L∞(L2)-norm, L2(In; H(div))-norm are derived. For pseudo hyperbolic problem, the optimal priori error estimates of the un-known function υ in L∞(H1)-norm and of auxiliary variables q＝auχ and σ＝υt-qχ in L∞(L2)-norm are obtained. Furthermore, a series of numerical experiments are provided to illustrate the effectiveness of the proposed methods.In the view of computing points, splitting mixed space-time discontinuous finite el-ement method takes full advantages of the two methods in numerical calculations. The subsystem of solving discrete auxiliary functions are separated from the coupled system of discrete displacement and auxiliary functions and the finite element approximation of the auxiliary functions can be solved by time discontinuous space-time finite element method with high accuracy. Therefore, the scale of the original problem and the difficulty of solving system are reduced to some extent and the mixed finite element spaces can also be selected flexibly.In Chapter8, We consider the space-time discontinuous finite element method, based on the equivalence integral equation, for nonlinear Sobolev equation. In this method, the equivalence integral equation is derived firstly by integrating the nonlinear Sobolev equa-tion with respect to t. Secondly, the equivalence integral equation is solved by time discontinuous space-time finite element method. This procedure has the following advan-tages:the convergence analysis is much simpler than that of the conventional space-time discontinuous finite element methods, because of precluding the time jump terms in the scheme. The discontinuity characteristics of approximate solution at time split points are embodied implicitly in the scheme. Moreover, the error estimate for time variable in L2(J)-norm can be derived directly. More importantly, this analysis can easily be extended to other initial boundary problems and nonlinear problems. Finally, some nu-merical results are given to illustrate the effectiveness of the proposed method.