高阶准亥姆霍兹投影算子：定义、分析与算法
High-order quasi-Helmholtz Projectors: Definition, Analyses, Algorithms
AI总结：
针对高阶离散电场积分方程的低频病态及标准准亥姆霍兹分解的缺陷，提出高阶准亥姆霍兹投影算子方法，无需识别拓扑环即可稳定方程，实现低频下恒定条件数，经理论与复杂场景数值验证有效。
摘要：
电场积分方程（EFIE）的精度可通过高阶离散化得到显著提升。然而，该方程在低频区间存在病态性和有害数值效应，往往会影响其求解。这一问题可通过准亥姆霍兹分解解决：将源单元和检验单元分解为无旋分量和有旋分量，再进行重新缩放，以同时避免低频条件数崩溃和数值精度损失。但标准准亥姆霍兹分解需要处理离散微分算子，这通常会加剧网格细化带来的病态性，还需要求解几何结构的拓扑环，在对复杂散射体建模（尤其是高阶情形下）时计算成本很高。本文通过首次将准亥姆霍兹投影算子推广到高阶离散化，并将其应用于高阶基函数离散的EFIE的稳定化，解决了上述缺陷。我们的策略无需识别拓扑环，且在频率降低时能保持条件数恒定。理论分析辅以数值结果，验证了该方法在复杂场景下的有效性。
英文摘要：
The accuracy of the electric field integral equation (EFIE) can be substantially improved using high-order discretizations. However, this equation suffers from ill-conditioning and deleterious numerical effects in the low-frequency regime, often jeopardizing its solution. This can be fixed using quasi-Helmholtz decompositions, in which the source and testing elements are separated into their solenoidal and non-solenoidal contributions, then rescaled in order to avoid both the low-frequency conditioning breakdown and the loss of numerical accuracy. However, standard quasi-Helmholtz decompositions require handling discretized differential operators that often worsen the mesh-refinement ill-conditioning and require the finding of the topological cycles of the geometry, which can be expensive when modeling complex scatterers, especially in high-order. This paper solves these drawbacks by presenting the first extension of the quasi-Helmholtz projectors to high-order discretizations and their application to the stabilization of the EFIE when discretized with high-order basis functions. Our strategy will not require the identification of the cycles and will provide constant condition numbers for decreasing frequencies. Theoretical considerations will be accompanied by numerical results showing the effectiveness of our method in complex scenarios.
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