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A marching-on-in-degree solver of time domain finite element-boundary integral method is proposed for the analysis of transient electromagnetic scattering from complex materials and fine detail structures. Unlike the marching-on-in-time solver of time domain finite element-boundary integral method, the proposed algorithm uses weighted Laguerre polynomials as temporal basis functions and testing functions, which leads to a recursive matrix equation solved degree by degree. Due to the property of weighted Laguerre polynomials, the proposed method does not involve late-time instability. Numerical results demonstrate that the proposed method is both stable and accurate.