Algebraic Multigrid Methods and Their Application to Generalized Finite Element Methods

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The research in this proposal is on the study and applications ofefficient algebraic multigrid methods for the solution of linearalgebraic systems arising from the discretization of second orderpartial differential equations by the generalized finite elementmethod. The proposed research will focus on the development andanalysis of adaptive techniques in the construction of hierarchy ofnested spaces and the choice of approximate subspace solvers that leadto the efficient and robust multigrid methods applicable to wide rangeof generalized finite element discretizations.The rapid increase in the power of today's supercomputers has made itfeasible for the scientific community to use numerical simulations tomodel physical phenomena to produce meaningful results. One of themodern techniques that can deliver quantitative results via suchsimulations is the generalized finite element method. This method hasproved to be a very robust discretization tool, applicable in variousbranches of engineering and sciences, for example, in simulating anddetermining the elastic, electromagnetic and other important physicalproperties of heterogeneous materials. Like most other discretizationtechniques, most often the majority of computation in such simulationsis devoted to the solution of the resulting linear systems ofequations. Hence, it is very important to develop efficient solversfor these systems. The results from the proposed research are thusexpected to have a broad and noticeable impact by providing the muchneeded iterative multilevel solution techniques for the discretelinear systems arising from numerical models in many applications.The proposed research is also expected to have an educational impactas it will provide a solid base for training of graduate students inthe modern theoretical and practical aspects of numerical methods forproblems in science and engineering.