Multiscale phenomena occur in a diverse range of science and engineering problems. For example, the vortical structures in the atmosphere may range from meters to thousands of kilometers. The time scales for the internal motion of proteins and nucleic acids typically span from 10^{-14} seconds to seconds. Other examples include moving contact line in immiscible flows and dynamics of microcrack in the fracture. These are fundamental problems of nature, whose solution/understanding is not only of interest to basic research, but also of importance to diverse applications in physics and engineering.

There are two difficulties inherent in an attack on singular problems with multiple scales. The first is that the governing physical laws at different scales are often different. The second is that computational-wise they demand huge computational resources. Recently the ever increasing performance of computers, coupled with tremendous advance in computational methodology, has opened new perspectives in modeling. It is now possible to derive instant predictions of some realistic model, to compare these with the experiment and to modify the model if possible. Mathematics develops notions as well as theoretical and numerical methods which make both a clean and efficient treatment of the model potentially possible. To study the multiscale phenomena via simulations, there is a need in developing even more efficient numerical methods based on adaptive grid refinement techniques that can resolve both the macro-scale as well as micro-scales (of molecular size) of the system. When the system involves multi-physics, the continuum calculations will have to be coupled with calculations based on quantum theory and molecular dynamics, so that the overall behavior can be uncovered and simulated.

As the multiscale modeling and computation are highly interdisciplinary, it requires close collaboration between researchers from different departments. The Croucher Lab is envisioned to serve as a platform for fruitful collaborations between faculties of different disciplines at HKUST, and between faculty members, students, postdocs and visitors. The Lab will focus on developing advanced numerical and analytic methods for effective treatment of singular, multiscale problems and on applications to specific projects listed below.