Conventional finite-element methods (FEMs) rely on an underlying tessellation to describe the geometry and the basis functions that are used to represent the unknown quantity. Alternatively, however, it is possible to represent both the geometry and basis as a set of points. This alternative scheme has been used extensively in solid mechanics to compute stress and strain distributions. This paper presents an adaptation of the scheme to the analysis of electromagnetic problems in both the static and quasi-static regimes. It validates the proposed model against both analytical solutions and benchmarked FEMs. The paper demonstrates the efficacy of the proposed method by applying it to a range of problems.