拟亥姆霍兹 Calderón 乘法预条件用于高阶全局多迹积分方程
Quasi-Helmholtz Calderón Multiplicative Preconditioning for Higher-Order Global Multi-Trace Integral Equations
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- Ghent University（根特大学）
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中文总结 AI 辅助
本文提出高阶全局多迹积分方程，并采用基于高阶拟亥姆霍兹投影子的 Calderón 乘法预条件，分离螺线管与非螺线管分量，避免病态矩阵求逆，实现低频稳定，迭代求解高效，数值实验验证了其在复合介质体散射问题中的有效性。
中文摘要 AI 辅助
本文针对复合物体对时谐电磁散射问题，提出了一种高阶全局多迹积分方程。该高阶多迹公式采用基于高阶拟亥姆霍兹投影子的 Calderón 乘法预条件器进行预条件处理。高阶拟亥姆霍兹投影子将基函数的螺线管分量与非螺线管分量分离。对亥姆霍兹分量的独立访问避免了直接求逆病态混合 Gram 矩阵。这使得无需细化网格或采用对偶基函数即可应用 Calderón 乘法预条件。此外，由于螺线管和非螺线管分量可分别重新缩放，该方法还实现了低频稳定化。高阶拟亥姆霍兹投影子通过迭代方式计算，使迭代求解器能够高效求解预条件后的矩阵系统，在几十次迭代内获得精确解。针对复合介质体进行了数值实验，证实了所提预条件器对于以高阶基函数离散的全局多迹公式在密集网格和极低频率下的有效性。
英文摘要
The paper presents a higher-order global multi-trace integral equation for time-harmonic electromagnetic scattering by composite objects. The higher-order multi-trace formulation is preconditioned with a Calderón multiplicative preconditioner using higher-order quasi-Helmholtz projectors. The higher-order quasi-Helmholtz projectors separate the solenoidal and non-solenoidal components of the basis functions. Separate access to the Helmholtz components circumvents the explicit inversion of an ill-conditioned mixed Gram matrix. This enables the application of Calderón multiplicative preconditioning without refining the mesh and resorting to dual basis functions. Furthermore, it enables low-frequency stabilization, as the solenoidal and non-solenoidal components can be rescaled individually. The higher-order quasi-Helmholtz projectors are computed iteratively, enabling iterative solvers to efficiently solve the preconditioned matrix system, yielding accurate solutions in a few dozen iterations. Numerical experiments are conducted for composite dielectric bodies, confirming the effectiveness of the proposed preconditioner for the global multi-trace formulation discretized with higher-order basis functions for dense meshes and at very low frequencies
