Abstract: In the 80's and 90's a great deal of effort was devoted to the design of a posteriori error estimators for a variety of PDE. These are computable quantities, depending on the discrete solution(s) and data, that can be used to assess the quality of the approximation and improve it adaptively. Despite their practical success, adaptive processes have been shown to converge, and to exhibit optimal complexity, only recently and just for linear elliptic PDE. This course presents this new theory and discusses extensions and open questions. A list of topics follows:

A posteriori error estimates for linear elliptic PDE Design and convergence of adaptive finite element methods Optimal complexity General elliptic operators and the Laplace-Beltrami operator Saddle point problems: Stokes system and mixed methods Pointwise error control for linear PDE Semilinear and nonlinear PDE Variational inequalities and free boundary estimation

The lectures are based mostly, but not entirely, on work by R.H. Nochetto and collaborators (E. Baensch, K. Mekchay, P. Morin, K. Siebert, A. Schmidt, A. Veeser).