Primer of adaptive finite element methods

Adaptive finite element methods (AFEM) are a fundamental numerical instrument in science and engineering to approximate partial differential equations. In the 1980s and 1990s a great deal of effort was devoted to the design of a posteriori error estimators, following the pioneering work of Babuska. These are computable quantities, depending on the discrete solution(s) and data, that can be used to assess the approximation quality and improve it adaptively. Despite their practical success, adaptive processes have been shown to converge, and to exhibit optimal cardinality, only recently for dimension d > 1 and for linear elliptic PDE. These series of lectures presents an up-to-date discussion of AFEM encompassing the derivation of upper and lower a posteriori error bounds for residual-type estimators, including a critical look at the role of oscillation, the design of AFEM and its basic properties, as well as a complete discussion of convergence, contraction property and quasi-optimal cardinality of AFEM.