Element-wise techniques based on SVD or QR Decomposition completely get rid of the usual ill-conditioning inherent to the plane-wave discretizations of the Ultra-Weak Variational Formulation (UWVF) for the Helmholtz equation. Associated preconditioning strategies lead to very low condition numbers of the corresponding linear system matrices, without any limitation on the number of plane waves per element. In addition, some of these procedures have the advantage of considerably reducing the size of the final system to be solved without altering the accuracy of the numerical solution.

中文翻译：

---

改善平面波超弱变分公式化条件的局部策略

基于SVD或QR分解的基于元素的技术完全摆脱了Helmholtz方程的超弱变分公式（UWVF）的平面波离散化所固有的常见病态。相关的预处理策略导致相应线性系统矩阵的条件数非常低，而对每个元素的平面波数没有任何限制。另外，这些程序中的一些程序具有在不改变数值解的精度的情况下大大减小要解决的最终系统的尺寸的优点。