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

DMS 0072462摘要：本提案的主题是发展（数值和分析）一类新的摄动方法，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扰动方法，用于近似BVP和FBP的解，该方法基于经典扰动理论的思想，适用于与精确可解几何有微小偏差的区域。 这些方法很有趣，因为它们快速，易于实现，并且无需重大修改即可转换为三维和更高的维度。 当然，这样的计划是有限的，其收敛域的范围，这可能是相当小的，并在许多感兴趣的BVP和FBP的事实，域是一个简单的几何形状的大扰动。 这一障碍已被有效地克服在最近的工作中，布鲁诺Reitich在设置的声学和电磁散射通过引入解析延拓技术，特别是使用帕德逼近。 当前一类微扰方法所面临的第二个挑战是，由于在其评估中发生的微妙取消，它们遭受数值病态问题。 这个缺点已经被PI Reitich克服了，对于计算拉普拉斯方程的Dirichlet-Neumann算子（DNO）的问题，通过简单地改变变量来简单地改变域。 PI建议通过开发一种通用的微扰方法来解决BVP和FBP，其中包括解析延拓和域平坦技术来扩展上述结果。 到目前为止，这些技术仅独立应用于BVP。 第一个目标是首先实现他们同时计算DNO（拉普拉斯方程）的边值问题，然后将这些方法扩展到一个真正的FBP（模拟一个理想流体的界面的运动）尺寸的情况下。 随后将调查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）它们的几何形状会给它们带来挑战。 随后，彻底重新调查的电磁和声散射的问题将完成域平坦化技术实现，以克服数值病态问题。 最后，将考虑通过DNO在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被定义在复杂的域上，这些域可能会或可能不会随时间演化。 这些问题，如散射的电磁辐射从一个粗糙的表面上的流体或表面波的演变，提出了严重的理论和计算困难的应用数学家和工程师。当问题的域是简单的（矩形，圆形等）该问题通常可以通过经典方法显式求解。 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小偏差的域。 人们已经提出了沿着这些路线的许多方法，但是除非非常小心，否则它们会导致近似值随着近似值的细化而实际上退化。 由PI Fernando Reitich开发的一种新技术避免了这些困难，并为复杂几何形状的问题估计提供了一种令人兴奋的新方法。 然而，挑战依然存在，这些挑战是本提案的主题。 一个挑战是扩展我们的新方法，小域偏差的问题，这是一个简单的几何大偏差。 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。

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.