Superpositions of plane waves are known to approximate well the solutions of the Helmholtz equation. Their use in discretizations is typical of Trefftz methods for Helmholtz problems, aiming to achieve high accuracy with a small number of degrees of freedom. However, Trefftz methods lead to ill-conditioned linear systems, and it is often impossible to obtain the desired accuracy in floating-point arithmetic. In this paper we show that a judicious choice of plane waves can ensure high-accuracy solutions in a numerically stable way, in spite of having to solve such ill-conditioned systems. Numerical accuracy of plane wave methods is linked not only to the approximation space, but also to the size of the coefficients in the plane wave expansion. We show that the use of plane waves can lead to exponentially large coefficients, regardless of the orientations and the number of plane waves, and this causes numerical instability. We prove that all Helmholtz fields are continuous superposition of evanescent plane waves, i.e., plane waves with complex propagation vectors associated with exponential decay, and show that this leads to bounded representations. We provide a constructive scheme to select a set of real and complex-valued propagation vectors numerically. This results in an explicit selection of plane waves and an associated Trefftz method that achieves accuracy and stability. The theoretical analysis is provided for a two-dimensional domain with circular shape. However, the principles are general and we conclude the paper with a numerical experiment demonstrating practical applicability also for polygonal domains.

翻译：平面波叠加能够很好地逼近亥姆霍兹方程的解。这类方法在离散化中的应用是Trefftz方法处理亥姆霍兹问题的典型特征，旨在以较少的自由度获得高精度。然而，Trefftz方法会导致病态线性系统，并且通常无法在浮点运算中达到预期的精度。本文证明，尽管必须求解此类病态系统，但通过合理选择平面波，仍可在数值稳定的前提下获得高精度解。平面波方法的数值精度不仅与逼近空间有关，还与平面波展开中系数的量级相关。研究表明，无论平面波的方向和数量如何，其使用都可能导致指数级增长的系数，从而引发数值不稳定性。我们证明了所有亥姆霍兹场都是倏逝平面波（即具有与指数衰减相关的复传播矢量的平面波）的连续叠加，并证实这种表示形式具有有界性。我们提出了一种构造性方案，用于在数值上选择一组实值和复值传播矢量。由此得到明确的平面波选择方案及相应的Trefftz方法，实现了精度与稳定性的统一。理论分析针对二维圆形区域展开，但所提原理具有普适性。文章末尾通过数值实验证明了该方法在多边形区域中的实际适用性。