发表机构
University of Arizona（亚利桑那大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文针对一阶时谐Maxwell方程，确定最优拟Trefftz空间并构造基，在保持高阶最佳逼近性质的同时显著降低维数，理论结合数值实验验证。
AI 中文摘要
本文旨在研究一阶时谐Maxwell方程的最优高阶Taylor基多项式拟Trefftz空间。与标准多项式空间相比，拟Trefftz空间的目的是在保持相同的高阶最佳逼近性质的同时提高效率：其维数显著更小。为实现这一目标，它们仅逼近控制方程的平滑解，而非逼近一般的平滑函数。虽然对于标量方程存在Taylor基多项式拟Trefftz空间的自然定义，但对于向量值方程，情况根本不同，因为朴素的拟Trefftz性质并不系统地定义具有期望最佳逼近性质的最小空间。本文确定了一阶时谐Maxwell方程的最优拟Trefftz空间，提出了两种构造拟Trefftz基的方法，并证明了这些空间的最佳逼近性质。这些理论结果强烈依赖于多项式Helmholtz分解和多项式de Rham序列的精确性，并通过数值实验进行了验证。
英文摘要
The goal of this article is to study optimal high-order Taylor-based polynomial quasi-Trefftz spaces for first-order time-harmonic Maxwell's equations. Compared with standard polynomial spaces, the purpose of quasi-Trefftz spaces is to retain the same high-order best approximation property but more efficiently: their dimension is considerably smaller. To achieve this goal, they approximate only smooth solutions to the governing equation rather than to approximate general smooth functions. While there is a natural definition of Taylor-based polynomial quasi-Trefftz spaces for scalar equations, the situation is fundamentally different for vector-valued equations, as the naive quasi-Trefftz property does not systematically define the smallest space with the desired best approximation property. This article identifies the optimal quasi-Trefftz spaces for first-order time-harmonic Maxwell's equations, proposes two approaches to construct quasi-Trefftz bases, and proves the spaces' best approximation property. The theoretical results, which strongly leverage polynomial Helmholtz decompositions and the exactness of the polynomial de Rham sequences, are validated with numerical experiments.
