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In a large variety of advanced, pulsed power systems with complex dielectric structures, there is a stringent need for evaluating the field distributions evolution in conditions of fast transients coupled with the nonuniformity and nonhomogeneity of dielectric materials-with a large spectrum of time-varying properties. The paper introduces the structures of three-dimensional finite element method (FEM) and boundary element method (BEM) numerical codes for analysis of such systems. The Galerkin method is at the basis of the formulation for the FEM applied for the lossy part of the system, while the BEM is applied to the lossless region. The numerical treatment has a special capability to isolate points and regions of critical phenomena such as corona or incipient breakdown and to regrid such regions of interest with a finer mesh using the coarse grid results to interpolate the boundary conditions. Several examples are presented for illustrative purposes.