Least-Squares Finite Element Methods and Optimization-Based Domain Decomposition Methods for Partial Differential Equations

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9806358 Gunzburger In the last few years, the engineering and mathematical communities have shown increasing interest in least-squares finite element methods for solving a variety of problems in fluids, electromagnetics, elasticity, and other applications. The great promise of least-squares methods arises from the fact that, when compared to other discretization schemes, they lead to discrete problems that are much easier to solve on a computer. In the past the PI has studied numerous facets of least-squares finite element methods. These include: the use of mesh-dependent weights in least-squares functionals in order to achieve optimally accuracy, the solution of practical implementation issues that needed to be addressed in order to make these methods practical and competitive, and the application of these methods to problems with discontinuous coefficients that arise, e.g., from inhomogeneous media properties. The PI plans to apply least-squares finite element methodologies to optimization and control problems and to develop, analyze, and implement domain decomposition algorithms in the least-squares setting. Domain decomposition methods have attracted even more attention due to their usefulness in a parallel processing environment. The PI has developed novel non-overlapping domain decomposition methods based on optimization or optimal control ideas that posses numerous desirable features, the most important perhaps being that they are easily extended to nonlinear problems. The PI plans to introduce preconditioners to speed-up the performance of the methods, to look at different functionals and optimization parameters on which to base the decomposition into subdomains so that again more efficient algorithms are obtained, to apply and analyze algorithms to the solution of optimization problems for partial differential equations, to develop algorithms for time-dependent problems, and to implement algorithms on parallel computers consisting of clusters of Pentium processors.