Improving Mixed Methods by Hybridization and Multigrid Techniques

该方案考虑了采用杂交和多重网格技术的创新数值方法。虽然杂交在一开始就被认为是一种实现技术，但研究者认为它远不止于此。它被证明是一种计算技术，在从计算流体力学到电磁学的应用中具有实现传统技术难以实现的目标的潜力。例如，提出了一种使用杂交的新技术来解决长期存在的研究问题，即精确计算流体流动的不可压缩近似。杂交的其他建议用途包括构建和分析可变程度混合方法以及设计局部后处理技术以提高准确性。另一类被提议的活动涉及多网格方法的加速解决技术。针对由计算电磁学和杂化引起的各种系统，提出了强大的多重网格技术。其目的是使解决方案技术有效。这一目标通常是通过多网格方法实现的，但也可以通过利用实际问题中出现的对称性（如轴对称）来实现。例如，研究轴对称麦克斯韦方程组的方法是拟议活动的一部分。虽然拟议的研究具有重要的理论影响，但同样重要的是，部分研究是由临床消融等工业应用需求驱动的。在这些应用中使用所提出的新方法是提案的一个组成部分。拟议活动的更广泛影响不仅来自对工业问题的应用，而且来自教育活动和成果的广泛传播。

This proposal considers innovative numerical methods using hybridization and multigrid techniques. While hybridization was thought of as an implementation technique in its inception, the investigator argues that it is much more. It is shown to be a computational technique that has the potential of achieving goals difficult to achieve by conventional techniques in applications ranging from computational fluid mechanics to electromagnetics. E.g., a new technique using hybridization to solve a long standing research problem, namely computing exactly incompressible approximations to fluid flow, is proposed. Other proposed uses of hybridization include construction and analysis of variable degree mixed methods and design of local post-processing techniques to enhance accuracy. A different category of proposed activities involve accelerated solution techniques by multigrid methods. Powerful multigrid techniques are proposed for various systems arising from computational electromagnetics and hybridization. The aim is to make solution techniques efficient. This goal, while often achieved by multigrid methods, can also be achieved by utilizing symmetries such as axisymmetry occurring in practical problems. E.g., investigation of methods for the axisymmetric Maxwell equations form a part of the proposed activities. While the proposed research has significant theoretical impact, equally important is the fact that parts of the research are motivated by the needs of industrial applications such as clinical ablation. Use of the proposed new methods in these applications is an integral part of the proposal. Broader impacts of the proposed activities result not only from the applications to the industrial problems, but also from educational activities and broad dissemination of results.