The proposed research is to design robust preconditioners for solving the finite element discretization of second order elliptic PDEs with variable coefficients and then apply these preconditioners in developing efficient iterative solvers for Maxwell's equations and H(div) equations. The PI shall develop, analyze and implement nearly optimal iterative solvers for elliptic PDE with variable coefficients, especially multiscale elliptic PDEs. Based on several successful preliminary investigations, the PI will design robust multilevel preconditioners for various types of finite element discretization, such as conforming, nonconforming, discontinuous Galerkin discretization and mixed formulation of second order elliptic PDEs with general variable coefficients on both structured and unstructured bisection grids. The approach will be based on the auxiliary space preconditioning framework. This technique will be proved to be robust with respect to both the variations in the coefficients and the grid size for solving general multiscale elliptic PDEs. The research will allow one to choose different coarse grid problems other than the standard variational coarse grid problems in the preconditioners. Upon obtaining robust preconditioners for various finite element discretizations, the PI will then use these preconditioners to develop robust iterative solvers for Maxwell's equations and H(div) equations with variable coefficients using the auxiliary space preconditioning techniques. The algorithms will be implemented as open source software packages which will be used in collaborations with domain-specific scientists.
This project has broad impact in education and other areas of mathematics, engineering, and physics through software development. The methods to be developed will contribute to the advancement of numerical methods for both linear and nonlinear systems. The research results will provide powerful tools for the exploration of important models such as reservoir simulations and electromagnetic computation. The project enriches the graduate program in the Department of Mathematics at ISU, especially in the area of partial differential equations and numerical analysis. The research shall involve undergraduate and graduate students, and excellent mentoring shall be provided by the PI.
