Discontinuous Galerkin Methods For Elliptic Variational Inequalities

In the past two decades, discontinuous Galerkin (DG) methods have been widely used for solving many kinds of mathematical and physical problems due to the flexibility in constructing feasible local shape function spaces and the ad-vantage to capture non-smooth or oscillatory solutions effectively. Nevertheless, there has been little work on DG methods for solving variational inequalities which form an important family of nonlinear problems arising in a wide variety of applications, such as elastoplasticity, contact mechanics, unilateral problems, heat control, etc. This Ph.D. thesis mainly focuses on the analysis of discontinu-ous Galerkin methods for solving elliptic variational inequalities, of both the first and second kinds.In Chapter 2, nine DG schemes with linear and quadratic elements for elliptic boundary value problems are extended to solve the obstacle problem, which is a representative elliptic variational inequality of the first kind. Consistency of these DG formulations is obtained and a priori error estimates are established, which reach optimal order for linear elements. Two numerical examples about LDG methods of solving the obstacle problem are reported to illustrate numerical convergence orders. Furthermore, in Chapter 3, a posteriori error analysis of the DG methods for solving the obstacle problem is studied. Error estimators of the residual type are proved to be reliable. Efficiency of the estimators is theoretically explored and numerically confirmed. Based on the a posteriori estimates analysis, an adaptive algorithm is proposed to solve the obstacle problem. At the end of this chapter, a numerical example is presented on adaptive LDG method for solving an obstacle problem over an L-shape domain. The results show that to obtain the same level of accuracy, comparing to the uniform refinement, adaptive strategy saves lots of time with smaller memory.For a simplified friction problem, which is an example of elliptic variational inequality of the second kind, featured by the presence of non-differentiable terms in the formulation, DG methods are studied in Chapter 4. Consistency of these DG schemes as well as boundedness and stability of the bilinear forms corre-sponding to certain norm (?)·(?)* are presented. Then a priori error estimates of these DG methods are analyzed and we obtain (?)u-uh(?)*≤Ch(p+1)/2, where the local polynomial degree p is an arbitrary nonnegative integer. Notice that when p=1, i.e., using the linear elements, these DG methods reach the optimal convergence order.In Chapter 5, DG methods are applied to solve the well-known Signorini problem, which is another example of elliptic variational inequality of the first kind. It is an elastostatics problem about finding the elastic equilibrium configu-ration of an elastic body contacting with a rigid frictionless surface. The process to derive general primal formulation of DG methods for the Signorini problem is presented. After making suitable choices of numerical fluxes, five consistent and stable DG formulations with linear elements are proposed. Unified error analysis for these DG schemes is given and the convergence orders are proved to be optimal. One numerical example is presented, and the results confirm the theoretical convergence order.In Chapter 6, three C0 DG methods for Kirchhoff plate problem are extended and two more C0 DG schemes are proposed to solve an elliptic variational in-equality of 4th-order. The conforming finite element method for solving 4th-order problems requires local polynomial degree p≥5, leading to much complication in construction and implementation of the method. So non-conforming finite elements are usually used instead. To solve the 4th-order problem, the C0 DG methods try to find the approximate solution in C0 continuous finite element spaces. Unlike fully discontinuous Galerkin methods, C0 type DG methods do not "double" the degrees of freedom at element boundaries. A priori error esti-mates are obtained for the C0 DG methods of the elliptic variational inequalities of 4th-order in unified way.