Polynomial Approximations of Singular Vector Fields

数学物理中的许多问题都需要在无界区域上近似解一个偏微分方程。例如计算三维空间中障碍物上的声波或电磁波的散射波。这类问题的数值处理是一个特别的挑战，因为底层区域必须用有限网格离散，如果区域是无界的，这并不是简单的。本项目的重点是边界单元法。其思想是离散化，而不是控制偏微分方程式，只存在于区域边界的相关积分方程式。这样，无界区域上的问题就归结为有界曲面上的相关问题(如果障碍是有界的)。在许多应用中，所考虑的区域或障碍都有角和边。上述问题的解决方案在那里表现不佳，出现了所谓的奇点。这些奇点极大地降低了数值逼近格式的性能。在这个项目中，我们研究了求解具有奇异性的波动问题(声学和电磁学)的数值格式，例如非光滑障碍物上的散射问题。对于波动问题，除了出现奇点之外，另一个困难是数值近似与精确(未知)解的相位差，即所谓的色散误差。我们建议使用高阶分段多项式来并行处理这两个问题：高阶多项式的奇异性对逼近性质的影响不那么严重，高阶多项式的数值色散误差比低阶多项式更有效地减少数值色散误差。目前几乎没有可用的数学理论来使用高阶方法来处理奇异波问题。这个项目的目的是提供这一理论，并发展有效地逼近波动问题奇异解的数值方法。一个特别的重点是分析和使用边界积分方程解无界域中的散射问题。

Many problems of mathematical physics require the approximate solution of a partial differential equation in an unbounded domain. Examples are the calculation of scattered acoustic or electromagnetic waves at an obstacle in the three dimensional space.The numerical treatment of such problems poses a particular challenge since the underlying domain has to be discretised by finite meshes and this is not straightforward if the domain is unbounded. Main focus in this project is on the boundary element method. The idea is to discretise, instead of the governing partial differential equation, a related integral equation that lives only on the boundary of the domain. In this way, the problem on an unbounded domain is reduced to a related problem on a bounded surface (if the obstacle is bounded).In many applications, the domain or obstacle under consideration has corners and edges. Solutions to the above mentioned problems are ill-behaved there, so-called singularities appear. These singularities greatly reduce the performance of numerical approximation schemes. In this project we study numerical schemes for the solution of wave problems (in acoustics and electromagnetism) with singularities, e.g. scattering problems at non-smooth obstacles. The additional difficulty with wave problems, apart from appearing singularities, is that numerical approximations suffer from a phase difference with the exact (unknown) solution, the so-called dispersion error. We propose to use high order piecewise polynomials for the approximation to tackle both problems in parallel: reduction of approximation properties by singularities are less severe for high order polynomials and numerical dispersion errors are much more efficiently reduced by high order polynomials than by those of low order.There is almost no mathematical theory available for the use of high order methods to deal with singular wave problems. The aim of this project is to provide this theory and to develop numerical methods that efficiently approximate singular solutions to wave problems. A particular focus is on the analysis and use of boundary integral equations for the solution of scattering problems in unbounded domains.