AI 中文总结
本文重访PWDG-UWVF等价性，通过结构解释明确其本质是同一边耦合问题的破场与阻抗迹坐标关系，将相关算子化为标准型，完善了该等价性的理论阐释。
AI 中文摘要
超弱变分形式（UWVF）是已知的Trefftz间断Galerkin格式，由标准PWDG通量族通过令α=β=δ=1/2得到。我们对该事实给出简短的结构解释：在每个内部网格边上，该参数值下的两项PWDG跳变项恰好是UWVF所用的两个定向阻抗传输缺陷平方范数之和的一半，因此Trefftz-DG范数是全局传输缺陷的自然Riesz范数。在PWDG坐标下，其Gram矩阵是离散矩阵的反Hermitian部分，Riesz归一化将算子化为标准型K+iiI（K为Hermitian矩阵）；将同一论证直接应用于离散UWVF，则得到互补标准型½I+iiK_U。这表明经典等价性是同一边耦合问题的破场坐标与阻抗迹坐标之间的关系，而非仅为惩罚项的特定选择。
英文摘要
The ultra-weak variational formulation (UWVF) is known to be the Trefftz discontinuous Galerkin scheme obtained from the standard PWDG flux family by setting \(α=β=δ=1/2\). We give a short structural explanation of this fact. On every interior mesh edge, the two PWDG jump terms at these parameter values are exactly one half of the sum of the squared norms of the two oriented impedance-transmission defects used by the UWVF. Thus the Trefftz-DG norm is the natural Riesz norm of the global transmission defect. In PWDG coordinates its Gramian is the anti-Hermitian part of the discrete matrix, and Riesz normalization puts the operator in the normal form \(K+\ii I\), with \(K\) Hermitian. The same argument applied directly to the discrete UWVF yields the complementary normal form \(\tfrac12 I+\ii K_{\rm U}\). This identifies the classical equivalence as a relation between broken-field and impedance-trace coordinates for the same edge-coupling problem, rather than only as a particular choice of penalties.
