AI 中文总结
本文针对二维标量电磁问题，提出梯度增强型FLAME、协调Trefftz-FLAME有限元等新Trefftz近似方法，经波散射算例验证，GEFLAME精度优于二次有限元，可应用于静电学、周期结构等场景。
AI 中文摘要
Trefftz函数可局部且精确满足微分方程，拟Trefftz函数则按规定的高阶近似满足该方程。本文研究二维标量电磁问题的（拟）Trefftz近似，已有的离散化方法包括灵活局部近似方法（Flexible Local Approximation MEthod, FLAME）、Trefftz单元（$T$-elements）以及Trefftz间断Galerkin（Trefftz-DG）方法。新进展包括梯度增强型FLAME（GEFLAME）、协调Trefftz-FLAME有限元（TFF），以及结合Schur-DtN约化的全场布洛赫波矢-频率解。应用范围涵盖静电学、散射、奇异场和周期结构中的布洛赫波。这些方法在协调性与灵活性间存在不同权衡：FLAME和GEFLAME将Trefftz函数直接融入局部差分格式；TFF采用单元级FLAME格式将多项式迹提升至单元内部；$T$-elements以弱形式匹配单元级Trefftz空间与此类迹；Trefftz-DG则通过通量和罚项耦合间断Trefftz空间。在已报道的波散射算例中，GEFLAME在相当的代数计算成本下，场和梯度误差比二次有限元离散化低数个数量级；局域奇异函数富集消除了主导的 reentrant-corner 误差。协调TFF和拟协调$T$-elements可采用标准有限元组装，但会暴露多项式边迹作为精度瓶颈。对于周期介质，双线性Trefftz-DG公式给出解析多项式波数-频率本征问题，且不限制布洛赫乘子在单位圆内；Schur-DtN约化得到紧凑的边界响应问题。
英文摘要
Trefftz functions satisfy the differential equation locally and exactly; quasi-Trefftz functions do so approximately to prescribed high order. This paper considers (quasi-)Trefftz approximations for two-dimensional scalar electromagnetic problems. Established discretizations include the Flexible Local Approximation MEthod (FLAME), Trefftz elements ($T$-elements), and Trefftz discontinuous Galerkin (Trefftz-DG) methods. New developments are gradient-enriched FLAME (GEFLAME), conforming Trefftz--FLAME finite elements (TFF), and a full-field Bloch-wavevector-vs-frequency solution with Schur--DtN reduction. Applications cover electrostatics, scattering, singular fields, and Bloch waves in periodic structures. These methods make different compromises between conformity and flexibility. FLAME and GEFLAME incorporate Trefftz functions directly into local difference schemes; TFF uses elementwise FLAME schemes to lift polynomial traces into element interiors; $T$-elements match elementwise Trefftz spaces weakly to such traces; and Trefftz-DG couples broken Trefftz spaces through fluxes and penalties. In the reported wave-scattering examples, GEFLAME gives field and gradient errors several orders of magnitude below those of the quadratic finite-element discretization at comparable algebraic cost. Localized singular-function enrichment removes the dominant reentrant-corner error. Conforming TFF and quasi-conforming $T$-elements admit standard finite-element assembly but expose polynomial edge traces as an accuracy bottleneck. For periodic media, the bilinear Trefftz-DG formulation gives an analytic polynomial wavenumber-vs-frequency eigenproblem without restricting the Bloch multiplier to the unit circle. Schur--DtN reduction yields compact boundary-response problems.
Comments42 pages, 14 figures
