A Posteriori Error Estimators and Adaptivity for Finite Element Approximation of the Non-Homogeneous Dirichlet Problem

Techniques are developed for a posteriori error analysis of the non-homogeneous Dirichlet problem for the Laplacian giving computable error bounds for the error measured in the energy norm. The techniques are based on the equilibrated residual method that has proved to be reliable and accurate for the treatment of problems with homogeneous Dirichlet data. It is shown how the equilibrated residual method must be modified to include the practically important case of non-homogeneous Dirichlet data. Explicit and implicit a posteriori error estimators are derived and shown to be efficient and reliable. Numerical examples are provided illustrating the theory.     

Tides

An elementary analysis of the problem of tides on an ideal Earth from a point of view of the forces involved proves to be a valuable exercise in terms of the applied mathematics as well as the physics. The approximations show their value both for obtaining analytic solutions and for exhibiting the symmetry‐breaking at higher order.     

Water purification by membranes: The role of polymer science

Two of the greatest challenges facing the 21st century involve providing sustainable supplies of clean water and energy, two highly interrelated resources, at affordable costs. Membrane technology is expected to continue to dominate the water purification technologies owing to its energy efficiency. However, there is a need for improved membranes that have higher flux, are more selective, are less prone to various types of fouling, and are more resistant to the chemical environment, especially chlorine, of these processes. This article summarizes the nature of the global water problem and reviews the state of the art of membrane technology. Existing deficiencies of current membranes and the opportunities to resolve them with innovative polymer chemistry and physics are identified. Extensive background is provided to help the reader understand the fundamental issues involved. © 2010 Wiley Periodicals, Inc. J Polym Sci Part B: Polym Phys, 2010