An Efficient Spectral Approximation via Decoupling and Dimension Reduction for the Curl-Curl Problem

Abstract We propose an efficient spectral approximation method for the curl-curl problem in a spherical domain based on decoupling and dimension reduction. By introducing a Lagrange multiplier to control the divergence of the field, the original formulation is transformed into a mixed system that effectively suppresses numerical pollution. By employing vector spherical harmonic expansions and separation of variables, the mixed system is further decoupled into a sequence of independent one-dimensional problems associated with transverse electric (TE) and transverse magnetic (TM) modes, resulting in substantial computational savings. Rigorous error estimates and a complete algorithmic implementation are developed for the TE modes, while an algorithmic framework is provided for the strongly coupled TM modes. Numerical experiments are presented to verify the convergence of the proposed method and support the theoretical analysis.