In this paper, the nodal domain decomposition with relaxation (NDDR) scheme is proposed to solve the nonlinear finite-element (FE) problem in electromagnetic apparatus without assembling the global system of equations. Each sub-domain contains only one node with unknown magnetic vector potential, and the calculation of each sub-domain can be massively parallelized to utilize the prevalent parallel computing architectures. The sub-domain solver has excellent modularity for single instruction multiple data programming with a specific data structure, and the required memory shows a linear increase with the problem size. The NDDR scheme is implemented on both multi core CPUs and many-core GPUs, and the accuracy and efficiency are discussed for different problem sizes. Result comparison with a commercial FE package shows a speedup of more than 30 times for a magnetostatic case and an average speedup of more than 53 times for a time domain nonlinear FE case with different time steps while maintaining an error of less than 0.85%.