Robust Least Squares Discretization for Mixed Variational Formulations

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This project aims to enable more reliable, fast, and accurate simulation methods for a variety of applications in science and engineering, such as electromagnetism, elasticity and acoustics, fluid flow, and diffusion through heterogenous porous media. The project focuses on efficient computational methods based on finite element analysis. Specific applications include modeling compressible gas dynamics and computational fluid mechanics for atmospheric prediction and ocean fluid flow behavior, as well as computational solution of the time-harmonic Maxwell model with applications in nano-optics and analog signal packages. This project will provide interdisciplinary applied mathematics training and research experiences for students.The project will develop, analyze, and implement robust and efficient numerical algorithms for solving partial differential equations (PDEs) that admit variational formulations with different types of test and trial spaces. When approximating the solutions of these PDEs, it is desirable to obtain robust estimates of all physical quantities in the presence of parameters, such as diffusion coefficient or frequency. It is also important to obtain good approximations of the solution, even in the case of low regularity near boundaries or along material discontinuities, or in the case of low data regularity. The focus of the project is on approximating PDE models with parameters and discontinuous coefficients. The project develops a general discretization and algorithm development method that bridges between the field of symmetric saddle point problems and the field of preconditioning elliptic symmetric problems. The project aims to: introduce a new saddle point least-squares theory that allows non-conforming trial spaces to approximate discontinuous solutions of PDEs; construct locally smooth projection type of trial spaces that lead to higher order of approximation for the solution or related quantities of interest; introduce optimal test spaces and balanced norms that allow robust approximation for parametric problems; and construct efficient preconditioning techniques for general mixed variational formulations. The research will broaden the mathematical theory and the range of applications of the mixed finite-element approximation field and will create new connections among mixed variational formulations, adaptive and multilevel techniques, and preconditioning parametric or fractional norms.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.