Negative Norm Least-Squares Finite Element Methods for Electromagnetics

9805590张本项目的目标是设计和分析的强大和准确的有限元方法所产生的问题，从力学和电磁学和迭代算法的建设，以解决由此产生的代数系统。 本研究将集中于探讨div-curl系统与麦克斯韦方程组的负范数最小二乘有限元方法。 这些方程代表了一类由流体和电磁学引起的一阶系统。 所提出的工作的应用发生在，例如，加速器和束流控制磁铁的设计，聚变装置，和电机design.Finite元素方法的基础上负范数最小二乘原理提供了一个自然的方法来数值求解这些问题。 负范数最小二乘方法有许多吸引人的特点。 像混合有限元方法一样，最感兴趣的变量可以直接近似而不是后验。 然而，最小二乘方法不受inf-sup稳定性条件的约束，因此仅根据近似性质和计算成本来选择有限元空间。 负范数最小二乘方法的收敛性只需要解的适度光滑。 此外，相应的代数系统是对称和正定的，因此可以通过例如，预条件共轭梯度法 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性。

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.