New Approaches in Solving Saddle Point Problems

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This research project provides a plan for the development, analysis, and implementation of faster and more efficient computational methods for partial differential equations (PDEs), including second-order elliptic problems, Stokes and Navier-Stokes systems, elasticity problems, and Maxwell's equations. PDEs have applications to many fields such as mathematics, physics, chemistry, biology, and engineering. The new methods for solving Stokes and Navier-Stokes systems will contribute to modeling flow problems of increasing size and complexity, e.g., modeling the air flow near a plane or modeling the flow near immersible objects. The new approach for Maxwell's equations has practical applications to photonics in computing modes for devices and in radar scattering. The research will focus on two areas: solving saddle-point problems and discretization on non-matching grids. To build new and efficient algorithms, the PI will combine his new ideas on solving saddle-point problems with already known methods from distinctive fields of numerical analysis such as iterative methods, multilevel methods, and adaptive methods for elliptic PDEs. The main technique will be based on the new spectral results for saddle-point systems found by the PI. The proposed work for solving saddle-point systems has scientific and technical applications in optimization, optimal control, computational fluid dynamics, linear elasticity, electromagnetism, electrical networks, linear models in statistics, and image restoration. A second area of research will be to investigate multilevel discretizations and multilevel preconditioning techniques in the context of discretizations on nonmatching grids based on the Partition of Unity method. The applications are to modeling fluid and gas flow near complex objects and fluid flow in porous media.