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 0072462ABSTRATCT。该提案的主题是一种新的扰动方法的开发（数值和分析），用于估计数学物理学中产生的边界价值问题解决方案（BVP）和自由边界问题（FBP）。 奥斯卡·布鲁诺（Oscar Bruno）和费尔南多·里蒂奇（Fernando Reitich）最近提出了一种新的扰动方法，用于近似于BVP和FBP的解决方案，该方法基于应用于域的经典扰动理论的思想，这些理论是与准确溶解的几何形状相比的小偏差。 这些方法很有趣，因为它们快速，易于实现，并转化为三个和更高的维度，而没有重大修改。 当然，这样的方案受其融合域的程度的限制，这可能很小，而且在许多bvp和FBP中，该域是一个简单几何形状的巨大扰动。 在Bruno＆Reitich最近通过引入分析延续技术（尤其是使用幻象近似因素）来设置声学和电磁散射的工作中，这种障碍已被有效克服。 当前类扰动方法所面临的第二个挑战是，由于其评估中发生的微妙取消，它们遭受了数值不良条件的问题。 Pi＆Reitich已经克服了这一缺点，因为它通过直接变化的变量来计算Dirichlet-Neumann运算符（DNO）用于Laplace方程的问题，而变量只是简单地弄平了域。 PI建议通过开发一种通用扰动方法来解决上述结果，以解决BVP和FBP，该方法结合了分析延续和域变平技术。 迄今为止，这些技术仅独立应用于BVP。 第一个目的是首先针对计算DNO的BVP同时实施它们（对于Laplace的方程式），然后将这些方法扩展到真正的FBP（建模理想流体的界面运动）维度的情况下。 将随后对底部地形和多个流体层对方法的影响进行调查，并将考虑其他经典FBP（例如Hele-Shaw流量，Stefan问题等）的考虑，其几何形状将带来自己的挑战。 随后，将通过实施的域扁平技术来完成对电磁和声学散射问题的彻底重新调查，以克服数值不良条件问题。 最后，将考虑通过DNO实施透明边界条件在散射问题中实施透明边界条件的问题。许多重要的科学问题是在可能随着时间推移或可能不会发展的复杂领域定义的。 这些问题，例如电磁辐射从粗糙的表面散射，或在流体上的表面波的演变，对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，问题通常可以通过经典方法明确解决。 估计更一般问题的一种方法是首先考虑与简单几何形状相比的小偏差的域。 已经提出了许多沿着这些线路的方法，但是除非非常小心，否则它们可能会导致近似值实际上降级，因为近似值被完善。 Pi＆Fernando Reitich开发的一种新技术避免了这种困难，并为估计复杂几何形状问题的问题提供了一种令人兴奋的新方法。 但是，挑战仍然存在，这些是该提议的主题。 一个挑战是将我们的小域偏差的新方法扩展到与简单几何形状的大偏差的问题。 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施这些新方法。