This book is a snapshot of current research in multiscale modeling, computations and applications. It covers fundamental mathematical theory, numerical algorithms as well as practical computational advice for analysing single and multiphysics models containing a variety of scales in time and space. Complex fluids, porous media flow and oscillatory dynamical systems are treated in some extra depth, as well as tools like analytical and numerical homogenization, and fast multipole method.

This book is a snapshot of current research in multiscale modeling, computations and applications. It covers fundamental mathematical theory, numerical algorithms as well as practical computational advice for analysing single and multiphysics models containing a variety of scales in time and space. Complex fluids, porous media flow and oscillatory dynamical systems are treated in some extra depth, as well as tools like analytical and numerical homogenization, and fast multipole method.

Multiscale problems naturally pose severe challenges for computational science and engineering. The smaller scales must be well resolved over the range of the larger scales. Challenging multiscale problems are very common and are found in e.g. materials science, fluid mechanics, electrical and mechanical engineering. Homogenization, subgrid modelling, heterogeneous multiscale methods, multigrid, multipole, and adaptive algorithms are examples of methods to tackle these problems. This volume is an overview of current mathematical and computational methods for problems with multiple scales with applications in chemistry, physics and engineering.

It is now 30 years since the network for digital communication, the ARPA-net, first came into operation. Since the first experiments with sending electronic mail and performing file transfers, the development of networks has been truly remarkable. Today's Internet continues to develop at an exponential rate that even surpasses that of computing and storage technologies. About five years after being commercialized, it has become as pervasive as the tele phone had become 30 years after its initial deployment. In the United States, the size of the Internet industry already exceeds that of the auto industry, which has been in existence for about 100 years. The exponentially increas ing capabilities of communication, computing, and storage systems is also reshaping the way science and engineering are pursued. Large-scale simulation studies in chemistry, physics, engineering, and sev eral other disciplines may now produce data sets of ,several terabytes or petabytes. Similarly, almost all measurements today produce data in digital form, whether from collections of sensors, three-dimensional digital images, or video. These data sets often represent complex phenomena that require rich visualization capabilities and efficient data-mining techniques to under stand. Furthermore, the data may be produced and archived in several differ ent locations, and the analysis carried out by teams with members at several locations-possibly distinct from those with significant storage, computation, or visualization facilities. The emerging computational Grids enable the transparent use of remote instruments, computational and data resources.

Most problems in science involve many scales in time and space. An example is turbulent ?ow where the important large scale quantities of lift and drag of a wing depend on the behavior of the small vortices in the boundarylayer. Another example is chemical reactions with concentrations of the species varying over seconds and hours while the time scale of the oscillations of the chemical bonds is of the order of femtoseconds. A third example from structural mechanics is the stress and strain in a solid beam which is well described by macroscopic equations but at the tip of a crack modeling details on a microscale are needed. A common dif?culty with the simulation of these problems and many others in physics, chemistry and biology is that an attempt to represent all scales will lead to an enormous computational problem with unacceptably long computation times and large memory requirements. On the other hand, if the discretization at a coarse level ignoresthe?nescale informationthenthesolutionwillnotbephysicallymeaningful. The in?uence of the ?ne scales must be incorporated into the model. This volume is the result of a Summer School on Multiscale Modeling and S- ulation in Science held at Boso n, Lidingo outside Stockholm, Sweden, in June 2007. Sixty PhD students from applied mathematics, the sciences and engineering parti- pated in the summer school."