Discontinuous Galerkin Methods For Linear Elasticity

The linear elasticity problems frequently occur in the fields of civil engineering and architecture. It is both theoretically and practically important to investigate numeri-cal methods for such problems. This thesis is intended to devise general discontinuous Galerkin (DG) methods for solving the linear elasticity problems, which contain the important local discontinuous Galerkin (LDG) method as a special example. We es-tablish error analysis for these methods. In particular, for the LDG method, a reliable and efficient residual-based a posteriori error estimator is proposed, from which an adaptive discontinuous Galerkin method (ALDGM.) is designed and the convergence of the algorithm together with its numerical performance is thoroughly discussed.Firstly, the DG methods are constructed. Using the idea of numerical traces on the boundaries of elements, a general framework of constructing DG methods is developed for solving the linear elasticity problems. The numerical traces are determined in view of a discrete stability identity, leading to a class of stable DG methods. A particular method, called the LDG method for linear elasticity, is studied in depth, which can be viewed as an extension of the LDG method of second order elliptic problems. The error bounds in L2-norm, H1-norm, and a certain broken energy norm are obtained. Some numerical results are provided to confirm the convergence theory established.Secondly, numerical solutions of the general form DG methods are proved to con-verge to the exact solutions. By defining two functionals KA and KB, the error es-timates in a certain seminorm|(·,·)| is shown. And then the error bounds in this seminorm and L2 norm are derived. Some numerical results are included to confirm our theoretical convergence orders.Thirdly, with some modification of the primal formulation in chapter 3, we achieve a new primal formulation ensuring the Galerkin orthogonality. With the help of this formulation, a residual-based a posteriori error estimator is proposed, and its reliability and efficiency are derived by means of a technical construction of uh and the approach of bubble functions.Finally, we design an adaptive local discontinuous Galerkin method(ALDGM) for the linear elasticity problems. Making use of the Galerkin orthogonality of the LDG method, we show the convergence of ALDGM. with respect to a quasi-error whenever the parameterηis big enough. The numerical examples are also performed to illustrate the convergence of our adaptive method.