Analysis of advanced discretizations of partial differential equations

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Modelling and simulation using partial differential equations are ubiquitous in science and engineering, including astronomy, meteorology, oceanography, seismology, geophysics, geology, economics, fluid mechanics, solid state physics, and quantum mechanics. In the last few decades we have seen an enormous increase in the complexity of a typical simulation that can be run on a computer. One reason is of course the amazing technological leaps in computer hardware we have been witnessing. On the other hand, perhaps surprisingly, it is estimated that the development of fast algorithms has had roughly the same impact as that of the hardware improvements. In fact, we expect much more: The existing fast algorithms and their theory can be compared to the tip of an iceberg, with most of the treasures yet to be discovered. The theme of the proposed project is the theoretical understanding of two important classes of numerical algorithms for solving partial differential equations: Adaptive methods and geometric discretization techniques. While adaptive methods can be described as algorithms that distribute computing resources in a "smartest" way so as to minimize waste of effort, the goal of geometric discretization techniques is to preserve fundamental geometric properties of the original differential equations. The latter techniques are known to be generally preferable to the more conventional methods, and in many cases, such as simulations of bio-chemical molecules, they are the only reasonable choice. The current project aims to further mathematical understanding of certain numerical algorithms belonging to either or both of the aforementioned classes. First, we plan to design new fast algorithms for solving various equations that are used to model bio-membranes, large scale ocean flows, and electromagnetic phenomena. Second, we will perform rigorous mathematical analysis of the designed algorithms, building a comprehensive theory that explains how the algorithms behave in different situations. This is important since no algorithm is completely fool proof, and depending on the particular problem at hand, one might want to choose different algorithms. The third aim of the project is mathematical analysis of some existing geometric discretization methods that are used in simulations of sub-nucleonic matter and of violent astronomical events such as black hole collisions. These methods work reasonably well in practice, but not without difficult unresolved problems, and currently we have a very little understanding of them from mathematical standpoint. We expect that rigorous mathematical treatment will not only give more confidence to the practitioners, but also open up possibilities to resolve the issues and improve upon the existing algorithms.