Convergence Analysis of the Spectral Element Method: A Comparative Study of Conformal Mappings for Eccentric Waveguide Modeling

Abstract

This paper compares the performance of two spectral element method mapping strategies for modeling electromagnetic fields in eccentric coaxial waveguides filled with uniaxially anisotropic media. A well-known cylindrical-based SEM (CSEM) mapping is compared with a novel conformal mapping that transforms the problem into an auxiliary rectangular domain, resulting in a rectangular SEM (RSEM) approach. Numerical results show that while both methods converge similarly for problems with large internal radii, the introduced RSEM offers faster convergence for small internal radii and offsets. Benchmarking against semi-analytical and finite element solutions demonstrates RSEM's superior efficiency and accuracy in solving problems in eccentric cylindrical domains with fewer degrees of freedom.