The author presents a general approach to a posteriori error estimation and adaptive mesh design for finite element models where the solution is subjected to inequality constraints. The local weighted residuals, that result from an extension of the so-called Dual-Weighted-Residual method, are used in a feed-back process for generating economical meshes. Based on several model problems, a general concept is proposed, which provides a systematic way of adaptive error control for problems stated in form of variational inequalities.

Abstract: "In this paper we are concerned with the numerical solution of stationary variational inequalities of obstacle type associated with second order elliptic differential operators in two or three space dimensions. In particular, we present adaptive finite element techniques featuring multilevel iterative solvers and a posteriori error estimators for local refinement of the triangulations. The algorithms rely on an outer-inner iterative scheme with an outer active set strategy and inner multilevel preconditioned cg-iterations involving variants of the hierarchical and the BPX-preconditioner which are derivded [sic] in the framework of multilevel additive Schwarz iterations. For the a posteriori error estimation in the energy norm three error estimators are presented which are based on the approximate solution of a quasivariational inequality satisfied by a piecewise quadratic approximation of the global discretization error. Finally, the performance of the preconditioners and the error estimators is illustrated by numerical results for a wide variety of stationary free boundary problems."

These Lecture Notes have been compiled from the material presented by the second author in a lecture series ('Nachdiplomvorlesung') at the Department of Mathematics of the ETH Zurich during the summer term 2002. Concepts of 'self adaptivity' in the numerical solution of differential equations are discussed with emphasis on Galerkin finite element methods. The key issues are a posteriori er ror estimation and automatic mesh adaptation. Besides the traditional approach of energy-norm error control, a new duality-based technique, the Dual Weighted Residual method (or shortly D WR method) for goal-oriented error estimation is discussed in detail. This method aims at economical computation of arbitrary quantities of physical interest by properly adapting the computational mesh. This is typically required in the design cycles of technical applications. For example, the drag coefficient of a body immersed in a viscous flow is computed, then it is minimized by varying certain control parameters, and finally the stability of the resulting flow is investigated by solving an eigenvalue problem. 'Goal-oriented' adaptivity is designed to achieve these tasks with minimal cost. The basics of the DWR method and various of its applications are described in the following survey articles: R. Rannacher [114], Error control in finite element computations. In: Proc. of Summer School Error Control and Adaptivity in Scientific Computing (H. Bulgak and C. Zenger, eds), pp. 247-278. Kluwer Academic Publishers, 1998. M. Braack and R. Rannacher [42], Adaptive finite element methods for low Mach-number flows with chemical reactions.

This is the third and yet further updated edition of a highly regarded mathematical text. Brenner develops the basic mathematical theory of the finite element method, the most widely used technique for engineering design and analysis. Her volume formalizes basic tools that are commonly used by researchers in the field but not previously published. The book is ideal for mathematicians as well as engineers and physical scientists. It can be used for a course that provides an introduction to basic functional analysis, approximation theory, and numerical analysis, while building upon and applying basic techniques of real variable theory. This new edition is substantially updated with additional exercises throughout and new chapters on Additive Schwarz Preconditioners and Adaptive Meshes.