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Nested multigrid techniques are combined with the ungauged vector and scalar potential formulation of the finite-element method to accelerate the convergence of the numerical solution of two-dimensional electromagnetic scattering problems. The finite-element modeling is performed on nested meshes of the same computational domain. The conjugate gradient method is used to solve the resultant finite-element matrix for the finest mesh, while the nested multigrid vector and scalar potential algorithm acts as the preconditioner of the iterative solver. Numerical experiments are used to demonstrate the superior numerical convergence and efficient memory usage of the proposed algorithm.