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A general and effective method is presented to numerically solve the electric field integral equation (EFIE) for topologically complex conducting domains by the finite element method. A new technique is proposed to decompose the surface current density into a solenoidal part and a nonsolenoidal remainder to avoid the low frequency breakdown. The surface current density field is approximated through div-conforming (facet) elements. The solenoidal part is represented through the space of the discrete approximation D of the surface divergence operator in the subspace spanned by the facet elements, whereas the nonsolenoidal remainder is represented through its complement. The basis functions of the space and its complement are evaluated, respectively, by the and pseudo-inverse of the matrix D. The completeness of the -pinv basis functions is studied. Unlike the loop-star and loop-tree basis functions, the -pinv basis functions allow to deal with topologically complex conducting domains in a general and readily applicable way. A topological interpretation of the "-pinv" decomposition is given and a general and simple method to evaluate the and pseudo-inverse of D is proposed. The computational complexity of the proposed method is discussed.