Adaptive Multigrid Methods for Partial Differential Equations

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ABSTRACT(NSF proposal: DMS-0074299, PI: Jinchao Xu)The proposed project is on the study of advanced solution methods for partial differential equations that arise from scientific and engineering applications. The theme of research is on the development, application and analysis of multigrid methods. The multigrid method is among the most powerful techniques for solving large scale linear and nonlinear systems arising from the discretization of partial differential equations. But the method has not been used in practice as often as they should be (because it is often not easy to code and to use), nor as efficiently as they could be (because it is often not easy to get the method work correctly). Our research is, on one hand, to develop special type of multigrid methods that are relatively easier to use for general users for some standard applications and is, on the other hand, to develop multigrid methods that are carefully tailored for some special class of practically interesting problems. One major component of the proposed research is a systematical investigation on various fundamental theoretical issues related to multigrid methods in general and also theoretical questions related to the algorithms to be developed for several specific problems of practical interests.The multigrid methods we propose to develop and study are expected to be applicable to a large class of practical problems including numerical simulations for electrochemical power devices (batteries) and advanced materials such as lattice block materials and liquid crystalline materials. These multigrid methods are expected to make a major impact in these and related applications and in particular to make it possible to simulate these problems in three dimensions in such a way that other traditional approaches may not be feasible. For example, the proposed study of advanced numerical methods for simulating electrochemical batteries has been and will be making a significant contribution to advanced battery technologies and manufactures that are vitally important in our everyday life, from the watch and the camera flash to electromobilies, modern space vehicles and wireless communications in information technology. Because of the practical backgrounds of these problems, the proposers and their research associates and graduate students are expected to actively interact and collaborate with physicists, engineers, computational scientists and practitioners from industries.