Numerical Methods for Elliptic Variational Inequalities in Higher Dimensions

Abstract

The main emphasis of this thesis is on developing and implementing linear and quadratic finite element methods for 3-dimensional (3D) elliptic obstacle problems. The study consists of a priori and a posteriori error analysis of conforming as well as discontinuous Galerkin methods on a 3D domain. The work in the thesis also focuses on constructing reliable and efficient error estimator for elliptic obstacle problem with inhomogeneous boundary data on a 2D domain. Finally, a MATLAB implementation of uniform mesh refinement for a 3D domain is also discussed. The first chapter is on introduction to the thesis, where we revisit some preliminary and fundamental results which will be used subsequently in the later chapters. We also present the literature survey and organization of the thesis. In Chapter 2, we present a quadratic finite element method for three dimensional elliptic obstacle problem which is optimally convergent (with respect to the regularity). We derive a priori error estimates to show the optimal convergence of the method with respect to the regularity, for this we have enriched the finite element space with element-wise bubble functions. Further, a posteriori error estimates are derived to design an adaptive mesh refinement algorithm. The chapter concludes by presenting a numerical experiment illustrating the theoretical result on a priori error estimate. Chapter 3 discusses two new discontinuous Galerkin (DG) finite element methods for the elliptic obstacle problem. Using the localized behaviour of DG methods, we derive a priori and a posteriori error estimates for linear and quadratic DG methods in dimension 2 and 3 without the addition of bubble functions. We consider two discrete sets, one with integral constraints (motivated from Chapter 2) and another with point constraints at quadrature points. The analysis is carried out in a unified setting which holds for several DG methods with variable polynomial degree. Later in Chapter 4, we propose a new and simpler residual based a posteriori error estimator for finite element approximation of the elliptic obstacle problem. The results in the chapter are two fold. Firstly, we address the influence of the inhomogeneous Dirichlet boundary condition in a posteriori error control of the elliptic obstacle problem. Secondly, by rewriting the obstacle problem in an equivalent form, we derive simpler a posteriori error bounds which are free from min/max functions. To accomplish this, we construct a post-processed solution ~uh of the discrete solution uh which satisfies the exact boundary conditions although the discrete solution uh may not satisfy. We propose two post processing methods and analyze them. We remark that the results known in the literature are either for the homogeneous Dirichlet boundary condition or that the estimator is only weakly reliable in the case of inhomogeneous Dirichlet boundary condition. In Chapter 5, we discuss uniform mesh refinement algorithm for a 3D domain. Starting with orientation of a face of the tetrahedron and orientation of the tetrahedron, we also discuss the ideas for nodes to element connectivity and red-refinement of a tetrahedron. Codes for the refinement are also included in this chapter. Finally, Chapter 6 concludes the thesis and motivates for possible extensions and also discusses about future works.