A grand challenge in computer simulations of complex systems is the reliability of computer predictions. A priori error estimates, as provided, e.g., by the standard a priori error analysis for finite element, finite volume, or finite difference methods, are often insufficient since they only yield information on the asymptotic error behavior and require regularity of the solution which are usually not satisfied by complex systems. Self-adaptive numerical methods such as Adaptive Mesh Refinement (AMR) algorithms provide a powerful and automatic approach to scientific computing.

The key ingredient for success of AMR algorithms is a posteriori error estimates that are able to accurately locate sources of global and local error in the current approximation. These considerations clearly show the need for an error estimator that can a posteriori be extracted from the computed numerical solution and the given data of the underlying problem. Such an error estimator is the so-called a posteriori error estimation that has been studied for the past four decades. In this talk, I will describe (1) basic principles of the a posteriori error estimation techniques for finite element approximations to partial differential equations and (2) our recent work.