Modeling of dispersion of air pollutants in the atmosphere is one of the most important and challenging scientific problems. There are several natural and anthropogenic events where passive or chemically active compounds are emitted into the atmosphere. The effect of these chemical species can have serious impacts on our environment and human health. Modeling the dispersion of air pollutants can predict this effect. Therefore, development of various model strategies is a key element for the governmental and scientific communities. We provide here a brief review on the mathematical modeling of the dispersion of air pollutants in the atmosphere. We discuss the advantages and drawbacks of several model tools and strategies, namely Gaussian, Lagrangian, Eulerian and CFD models. We especially focus on several recent advances in this multidisciplinary research field, like parallel computing using graphical processing units, or adaptive mesh refinement.

In the past years considerable attention has been devoted to designing and controlling patterns at the microscale using bottom-up self-assembling techniques. The precipitation process proved itself to be a good candidate for building complex structures. Therefore, the techniques and ideas to control the precipitation processes in space and in time play an important role. We present here a simple and technologically applicable technique to produce arbitrarily shaped precipitation (Liesegang) patterns. The precipitation process is modelled using a sol coagulation model, in which the precipitation occurs if the concentration of the intermediate species (sol) produced from the initially separated reactants (inner and outer electrolytes) reaches the coagulation threshold. Spatial and/or temporal variation of this threshold can result in equidistant and revert (inverse) type patterns in contrast to regular precipitation patterns, where during the pattern formation a constant coagulation threshold is supposed and applied in the simulations. In real systems, this threshold value may be controlled by parameters which directly affect it (e.g. temperature, light intensity or ionic strength).

An implicit a posteriori error estimation technique is presented and analyzed for the numerical solution of the time-harmonic Maxwell equations using Nédélec edge elements. For this purpose we define a weak formulation for the error on each element and provide an efficient and accurate numerical solution technique to solve the error equations locally. We investigate the well-posedness of the error equations and also consider the related eigenvalue problem for cubic elements. Numerical results for both smooth and non-smooth problems, including a problem with reentrant corners, show that an accurate prediction is obtained for the local error, and in particular the error distribution, which provides essential information to control an adaptation process. The error estimation technique is also compared with existing methods and provides significantly sharper estimates for a number of reported test cases.