Solutions of the Helmholtz equation are known to be well approximated by superpositions of propagative plane waves. This observation is the foundation of successful Trefftz methods. However, when too many plane waves are used, the computation of the expansion is known to be numerically unstable. We prove that exponentially large coefficients in the expansion are unavoidable and we explain how this causes instability. In this work, we show that the Helmholtz solutions on a disk can be exactly represented by a continuous superposition of evanescent plane waves, generalizing the standard Herglotz representation. Here, by evanescent plane waves, we mean exponential plane waves with complex-valued propagation vector, whose absolute value decays exponentially in one direction. In addition, the density in this representation is proved to be uniformly bounded in a suitable weighted Lebesgue norm, hence overcoming the instability observed with propagative plane waves and paving the way for stable discrete expansions. In view of practical implementations, discretization strategies are investigated. We construct suitable finite-dimensional sets of evanescent plane waves using sampling strategies in a parametric domain. Provided one uses sufficient oversampling and regularization, numerical experiments show that the resulting approximations are both controllably accurate and numerically stable.
翻译:Helmholtz 方程式的解决方案众所周知,其近似于传播性平面波的叠加位置。这一观测是成功的 Trefftz 方法的基础。然而,如果使用过多的平面波,则扩张的计算在数字上是不稳定的。我们证明扩张过程中的指数性高系数是不可避免的,我们解释这如何造成不稳定。在这项工作中,我们显示磁盘上的Helmholtz 方块溶液完全可以通过连续的双向双向双向双向双向双向双向双向双向双向双向双向双向。我们在这里,通过电子双向双向双向双向双向双向双向移动,其绝对值会在一个方向快速递减。此外,这一方位的密度被证明一致地捆绑在一个适当的加权偏斜的蓝方规范中,从而克服了以推进性平面飞机波观察到的不稳定性,并为稳定的离心扩张铺平路铺平路。为了实际的实施,我们调查了离心战略。我们使用一个准的取样策略,建造了适当的微维的日空波波。只要一个可靠的数字的实验,就能证明数字的正规化。