Negative Norm Least-Squares Finite Element Methods for Electromagnetics

该项目的目标是设计和分析由力学和电磁学引起的问题的鲁棒和精确的有限元方法，并构建求解所得到的代数系统的迭代算法。本研究将集中于负范数最小二乘有限元法的研究，并将其应用于复旋度系统和麦克斯韦方程组。这些方程是由流体和电磁学引起的一类一阶系统的代表。所提出的工作的应用发生在，例如，加速器和束控制磁体的设计，聚变装置，和电动机的设计。基于负范数最小二乘原理的有限元方法为这些问题的数值解提供了一种自然的方法。负范数最小二乘法有许多吸引人的特点。与混合有限元方法一样，最感兴趣的变量可以直接逼近，而不是事后逼近。然而，最小二乘法不受中稳定条件的约束，因此仅根据近似性质和计算成本来选择有限元空间。负范数最小二乘法的收敛性只要求解具有适度的平滑性。此外，所对应的代数系统是对称的、正定的，因此可以用预条件共轭梯度法等有效地求解。所提出的方法将利用模型问题的数学结构和负范数最小二乘原理的计算特点。

9805590ZhangThe 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.