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.
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- Mathematics (MA) [162]