High Order Boundary Perturbation Methods for Boundary Value and Free Boundary Problems

边界值和自由边界问题的高阶边界摄动方法

• 批准号: 0072462
• 负责人: David Nicholls
• 金额: $ 8.54万
• 依托单位: University of Minnesota-Twin Cities
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2000
• 资助国家: 美国
• 起止时间: 2000-07-15 至 2001-09-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0072462&HistoricalAwards=false
• 关键词: Order; Boundary; Perturbation; Methods; Value

DMS 0072462ABSTRACT.The subject of this proposal is the development (both numerical and analytical) of a new class of perturbative methods for estimating solutions of boundary value problems (BVP) and free boundary problems (FBP) arising in mathematical physics. Oscar Bruno and Fernando Reitich recently proposed a new class of perturbation methods for approximating solutions of BVP and FBP which are based on the ideas of classical perturbation theory applied to domains which are small deviations from exactly solvable geometries. These methods are interesting because they are fast, easy to implement, and translate into three and higher dimensions without major modification. Of course, such schemes are limited by the extent of their domain of convergence, which may be quite small, and the fact that in many BVP and FBP of interest, the domain is a large perturbation of a simple geometry. This obstacle has been effectively overcome in recent work of Bruno & Reitich in the setting of acoustic and electromagnetic scattering via the introduction of analytic continuation techniques, in particular the use of Pade approximants. A second challenge faced by the current class of perturbative methods is that they suffer from problems of numerical ill-conditioning due to subtle cancellations which take place in their evaluation. This drawback has been overcome by the PI & Reitich, for the problem of computing Dirichlet-Neumann operators (DNO) for Laplace's equation, via a straightforward change of variables which simply flattens the domain. The PI proposes to extend the above results by developing a general purpose perturbative method for solving BVP and FBP which incorporatesboth analytic continuation and domain flattening techniques. To date, the techniques have only been applied independently to BVP. The first objective is to first implement them simultaneously for the BVP of computing DNO (for Laplace's equation), and then extend these methods to the case of a genuine FBP (modeling the motion of the interface of an ideal fluid) dimensions. An investigation of the effects of bottom topography and multiple fluid layers on the methods will follow, and considerations will be made of other classical FBP (e.g. Hele-Shaw flows, Stefan problems, etc.) whose geometries will pose their own challenges. Subsequently a thorough re-investigation of the problems of electromagnetic and acoustic scattering will be completed with domain flattening techniques implemented to overcome numerical ill-conditioning problems. Finally, the problem of implementing transparent boundary conditions in scattering problems via DNO will be considered.Many important scientific problems are defined on complicated domains that may or may not evolve in time. These problems, such as the scattering of electromagnetic radiation from a rough surface or the evolution of surface waves on a fluid, pose severe theoretical and computational difficulties for applied mathematicians and engineers. When the domain of the problem is simple (rectangular, circular, etc.) the problem can usually be solved explicitly by classical methods. One approach to the estimation of more general problems is to first consider domains which are small deviations from simple geometries. Many approaches along these lines have been proposed, but unless great care is taken, they can result in approximations that actually degrade as the approximation is refined. A new technique, developed by the PI & Fernando Reitich, avoids such difficulties and provides an exciting new method for the estimation of problems on complicated geometries. However, challenges still remain and these are the subject of this proposal. One challenge is to extend our new methods for small domain deviations to problems which are large deviations from a simple geometry. Another challenge is the fast and efficient implementation of these new methods on high performance serial and parallel computers.

DMS 0072462文摘。本提案的主题是发展（数值和解析）一类新的微扰方法，用于估计数学物理中出现的边值问题（BVP）和自由边界问题（FBP）的解。Oscar Bruno和Fernando Reitich最近提出了一类新的微扰方法来逼近BVP和FBP的解，这些方法基于经典微扰理论的思想，应用于与精确可解几何的小偏差区域。这些方法很有趣，因为它们快速，易于实现，并且无需进行重大修改即可转换为三维或更高的维度。当然，这些方案受其收敛域范围的限制，可能相当小，而且在许多感兴趣的BVP和FBP中，该域是一个简单几何的大摄动。Bruno &amp； Reitich最近在声学和电磁散射设置中通过引入解析延拓技术，特别是使用Pade近似，有效地克服了这一障碍。当前一类微扰方法面临的第二个挑战是，由于在它们的计算中发生了微妙的取消，它们遭受了数值病态的问题。这个缺点已经被PI &amp； Reitich克服了，用于计算拉普拉斯方程的狄利克雷-诺伊曼算子（DNO）的问题，通过简单的变量变化使域平坦化。PI建议通过开发一种通用的微扰方法来扩展上述结果，该方法结合了解析延拓和区域平坦化技术来求解BVP和FBP。迄今为止，这些技术仅单独应用于BVP。第一个目标是首先将它们同时用于计算DNO（拉普拉斯方程）的BVP，然后将这些方法扩展到真正的FBP（模拟理想流体界面的运动）维度的情况下。接下来将研究底部地形和多流体层对方法的影响，并考虑其他经典的FBP（例如Hele-Shaw流，Stefan问题等），这些问题的几何形状将带来自己的挑战。随后，将对电磁散射和声散射问题进行彻底的重新研究，并采用域平坦化技术来克服数值病态问题。最后，讨论了在DNO散射问题中实现透明边界条件的问题。许多重要的科学问题都是在复杂的领域中定义的，这些领域可能会或可能不会随时间演变。这些问题，例如来自粗糙表面的电磁辐射散射或流体表面波的演变，给应用数学家和工程师带来了严重的理论和计算困难。当问题的域很简单（矩形、圆形等）时，通常可以用经典方法显式求解。估计更一般问题的一种方法是首先考虑与简单几何形状有微小偏差的域。沿着这条路线已经提出了许多方法，但是除非非常小心，否则它们可能导致随着近似的改进而实际上降级的近似。由PI &amp； Fernando Reitich开发的一项新技术避免了这些困难，并为复杂几何问题的估计提供了一种令人兴奋的新方法。然而，挑战仍然存在，这是本建议的主题。一个挑战是将我们的新方法扩展到与简单几何有很大偏差的问题上。另一个挑战是在高性能串行和并行计算机上快速有效地实现这些新方法。