Improving Mixed Methods by Hybridization and Multigrid Techniques

该提案考虑了使用杂交和多重网格技术的创新数值方法。虽然杂交最初被认为是一种实现技术，但研究人员辩称，它远不止如此。它被证明是一种计算技术，在从计算流体力学到电磁学的各种应用中，它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来解决一个长期研究问题的新技术，即计算流体流动的精确不可压缩近似。杂交的其他拟议用途包括构建和分析可变度混合方法，以及设计局部后处理技术以提高准确性。另一类拟议的活动涉及多重网格方法的加速求解技术。强大的多重网格技术被提出用于计算电磁学和杂化所产生的各种系统。其目的是使解决方案技术更有效。虽然这一目标通常是通过多重网格法实现的，但也可以通过利用实际问题中出现的轴对称等对称性来实现。例如，轴对称麦克斯韦方程方法的研究是拟议活动的一部分。虽然拟议的研究具有重大的理论影响，但同样重要的是，部分研究的动机是临床消融等工业应用的需求。在这些应用中使用拟议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源于对工业问题的应用，还源于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.