The main emphasis of this thesis is to study a posteriori error analysis of discontinuous Galerkin (DG) methods for the elliptic variational inequalities. The DG methods have become very pop-ular in the last two decades due to its nature of handling complex geometries, allowing irregular meshes with hanging nodes and different degrees of polynomial approximation on different ele-ments. Moreover they are high order accurate and stable methods. Adaptive algorithms refine the mesh locally in the region where the solution exhibits irregular behaviour and a posteriori error estimates are the main ingredients to steer the adaptive mesh refinement. The solution of linear elliptic problem exhibits singularities due to change in boundary con-ditions, irregularity of coefficients and reentrant corners in the domain. Apart from this, the solu-tion of variational inequality exhibits additional irregular behaviour due to occurrence of the free boundary (the part of the domain which is a priori unknown and must be found as a component of the solution). In the lack of full elliptic regularity of the solution, uniform refinement is inefficient and it does not yield optimal convergence rate. But adaptive refinement, which is based on the residuals ( or a posteriori error estimator) of the problem, enhance the efficiency by refining the mesh locally and provides the optimal convergence. In this thesis, we derive a posteriori error estimates of the DG methods for the elliptic variational inequalities of the first kind and the second kind. This thesis contains seven chapters including an introductory chapter and a concluding chap-ter. In the introductory chapter, we review some fundamental preliminary results which will be used in the subsequent analysis. In Chapter 2, a posteriori error estimates for a class of DG meth-ods have been derived for the second order elliptic obstacle problem, which is a prototype for elliptic variational inequalities of the first kind. The analysis of Chapter 2 is carried out for the general obstacle function therefore the error estimator obtained therein involves the min/max func-tion and hence the computation of the error estimator becomes a bit complicated. With a mild assumption on the trace of the obstacle, we have derived a significantly simple and easily com-putable error estimator in Chapter 3. Numerical experiments illustrates that this error estimator indeed behaves better than the error estimator derived in Chapter 2. In Chapter 4, we have carried out a posteriori analysis of DG methods for the Signorini problem which arises from the study of the frictionless contact problems. A nonlinear smoothing map from the DG finite element space to conforming finite element space has been constructed and used extensively, in the analysis of Chapter 2, Chapter 3 and Chapter 4. Also, a common property shared by all DG methods allows us to carry out the analysis in unified setting. In Chapter 5, we study the C0 interior penalty method for the plate frictional contact problem, which is a fourth order variational inequality of the second kind. In this chapter, we have also established the medius analysis along with a posteriori analy-sis. Numerical results have been presented at the end of every chapter to illustrate the theoretical results derived in respective chapters. We discuss the possible extension and future proposal of the work presented in the Chapter 6. In the last chapter, we have documented the FEM codes used in the numerical experiments.

Effiziente und zuverlässige a posteriori Fehlerabschätzungen sind eine Hauptzutat für die effiziente numerische Berechnung von Lösungen zu Variationsungleichungen durch die Finite-Elemente-Methode. Die vorliegende Arbeit untersucht zuverlässige und effiziente Fehlerabschätzungen für beliebige Finite-Elemente-Methoden und drei Variationsungleichungen, nämlich dem Hindernisproblem, dem Signorini Problem und dem Bingham Problem in zwei Raumdimensionen. Die Fehlerabschätzungen hängen vom zum Problem gehörenden Lagrange Multiplikator ab, der eine Verbindung zwischen der Variationsungleichung und dem zugehörigen linearen Problem darstellt. Effizienz und Zuverlässigkeit werden bezüglich eines totalen Fehlers gezeigt. Die Fehleranschätzungen fordern minimale Regularität. Die Approximation der exakten Lösung erfüllt die Dirichlet Randbedingungen und die Approximation des Lagrange Multiplikators ist nicht-positiv im Falle des Hindernis- und Signoriniproblems, und hat Betrag kleiner gleich 1 für das Bingham Problem. Dieses allgemeine Vorgehen ermöglicht das Einbinden nicht-exakter diskreter Lösungen, welche im Kontext dieser Ungleichungen auftreten. Aus dem Blickwinkel der Anwendungen ist Effizienz und Zuverlässigkeit im Bezug auf den Fehler der primalen Variablen in der Energienorm von großem Interesse. Solche Abschätzungen hängen von der Wahl eines effizienten diskreten Lagrange Multiplikators ab. Im Falle des Hindernis- und Signorini Problems werden postive Beispiele für drei Finite-Elemente Methoden, der konformen Courant Methode, der nicht-konformen Crouzeix-Raviart Methode und der gemischten Raviart-Thomas Methode niedrigster Ordnung hergeleitet. Partielle Resultate liegen im Fall des Bingham Problems vor. Numerischer Experimente heben die theoretischen Ergebnisse hervor und zeigen Effizienz und Zuverlässigkeit. Die numerischen Tests legen nahe, dass der aus den Abschätzungen resultierende adaptive Algorithmus mit optimaler Konvergenzrate konvergiert. / Efficient and reliable a posteriori error estimates are a key ingredient for the efficient numerical computation of solutions for variational inequalities by the finite element method. This thesis studies such reliable and efficient error estimates for arbitrary finite element methods and three representative variational inequalities, namely the obstacle problem, the Signorini problem, and the Bingham problem in two space dimensions. The error estimates rely on a problem connected Lagrange multiplier, which presents a connection between the variational inequality and the corresponding linear problem. Reliability and efficiency are shown with respect to some total error. Reliability and efficiency are shown under minimal regularity assumptions. The approximation to the exact solution satisfies the Dirichlet boundary conditions, and an approximation of the Lagrange multiplier is non-positive in the case of the obstacle and Signorini problem and has an absolute value smaller than 1 for the Bingham flow problem. These general assumptions allow for reliable and efficient a posteriori error analysis even in the presence of inexact solve, which naturally occurs in the context of variational inequalities. From the point of view of the applications, reliability and efficiency with respect to the error of the primal variable in the energy norm is of great interest. Such estimates depend on the efficient design of a discrete Lagrange multiplier. Affirmative examples of discrete Lagrange multipliers are presented for the obstacle and Signorini problem and three different first-order finite element methods, namely the conforming Courant, the non-conforming Crouzeix-Raviart, and the mixed Raviart-Thomas FEM. Partial results exist for the Bingham flow problem. Numerical experiments highlight the theoretical results, and show efficiency and reliability. The numerical tests suggest that the resulting adaptive algorithms converge with optimal convergence rates.