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This paper presents the application of a p-type multiplicative Schwarz (pMUS) method for solving three-dimensional waveguide discontinuity problems. The two major contributions of the proposed pMUS method are: 1) the use of hierarchical curl-conforming basis functions that incorporate a discrete Hodge decomposition explicitly and 2) the treatment of each polynomial space (or basis functions group) as an abstract grid/domain in the Schwarz method. These two features greatly improve the applicability of the curl-conforming vector finite-element methods (FEMs) for solving Maxwell equations. Various numerical examples are solved using the proposed approach. The performance of the pMUS method has been compared to commercial FEM software as well as the incomplete Choleski conjugate gradient method. It is found that the pMUS method exhibits superior efficiency and consumes far less memory and CPU times.