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We present a high‐accuracy numerical method based on a decoupled dimensionality reduction scheme for Maxwell eigenvalue problem in spherical domains. Using the orthogonality of vector spherical harmonics and the variable separation approach, we decompose the original problem into two classes of decoupled one‐dimensional TE mode and TM mode. For the TE mode, we establish a variational formulation and its discrete scheme and give the error estimations of the approximate eigenvalues and eigenfunctions. For the TM mode, it is different from TE mode which naturally meets the divergence‐free condition and will not generate some spurious eigenvalues. We design a numerical algorithm based on a parameterized method to filter out the spurious eigenvalues. Finally, some numerical results are presented to confirm the theoretical results and validate the algorithms. [ABSTRACT FROM AUTHOR]

In this paper, we propose and analyze an efficient spectral-Galerkin method based on a mixed formulation with dimension reduction for the Helmholtz transmission eigenvalue problem in spherical domains. By introducing an auxiliary function, we rewrite the original problem as an equivalent fourth-order coupled form in spherical coordinates. Using the properties of spherical harmonic and Laplace–Beltrami operator, we further decompose the original problem into a series of decoupled one-dimensional fourth-order linear eigenvalue problems, for which a new mixed variational formulation and its discretization is developed. For error estimates of numerical eigenvalues and eigenfunctions, we recall the spectral theory of compact operators. Towards this end, we derive the essential polar conditions, define a class of weighted Sobolev spaces, and most importantly, prove a sequence of two compact embedding properties for the weighted Sobolev spaces, based on which the spectral theory of compact operators for the variational formulation and discrete system can be established. Finally, some numerical examples are presented to confirm the theoretical error analysis and the efficiency of our algorithm. [ABSTRACT FROM AUTHOR]