The objectives of this project are the design and analysis of robust and accurate finite element methods for problems arising from mechanics and electromagnetics and the construction of iterative algorithms for solving the resulting algebraic systems. The research will focus on the investigation of negative norm least-squares finite element methods for div-curl systems and the Maxwell equations. These equations are representatives of a class of first order systems arising from fluid and electromagnetics. Applications of the proposed work occur in, for example, the design of accelerator and beam control magnets, fusion devices, and electrical motor design.
Finite element methods based on negative norm least-squares principles provide a natural approach to the numerical solution of those problems. The negative norm least-square methods have a number of attractive features. Like mixed finite element methods, variables of most interest can be approximated directly instead of a posteriori. However, least-squares methods are not subject to the inf-sup stability condition and hence the finite element spaces are selected based solely on approximation properties and computational cost. The convergence of negative norm least-squares methods requires only a moderate amount of smoothness on the solution. Moreover, the corresponding algebraic system is symmetric and positive definite and hence can be solved effectively by, e.g., the preconditioned conjugate gradient method. The proposed methods will exploit both the mathematical structure of model problems and the computational features of negative norm least-squares principles.
