{'fraction_non_alphanumeric': 0.03177570093457944, 'fraction_numerical': 0.009345794392523364, 'mean_word_length': 5.12, 'pattern_counts': {'":': 0, '<': 0, '<?xml version=': 0, '>': 0, 'https://': 0, 'lorem ipsum': 0, 'www.': 0, 'xml': 0}, 'pii_count': 0, 'substrings_counts': 0, 'word_list_counts': {'cursed_substrings.json': 0, 'profanity_word_list.json': 0, 'sexual_word_list.json': 0, 'zh_pornsignals.json': 0}}
Optimized Schwarz Method with Complete Radiation Transmission Conditions for the Helmholtz Equation in Waveguides

In this paper, we present a nonoverlapping optimized Schwarz algorithm with a high-order transmission condition for the Helmholtz equation posed in waveguides. We introduce the new high-order transmission conditions based on the complete radiation boundary conditions (CRBCs) that have been developed for high-order absorbing boundary conditions. We obtain the convergence rate of the algorithm in terms of reflection coefficients of CRBCs, which decrease exponentially with the order of CRBCs. It will be shown that damping parameters involved in the transmission conditions can be selected in an optimal way for enhancing the convergence of the Schwarz algorithm. Also, this algorithm can be employed efficiently for a preconditioner in GMRES implementations based on the substructured form as in [M. J. Gander, F. Magoules, and F. Nataf, SIAM J. Sci. Comput., 24 (2002), pp. 38--60]. Finally, numerical examples confirming the theory will be presented.