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The Galerkin method with subsectional linear basis and weighting functions is used to calculate the low-frequency transverse electric fields in lossy dielectric cylinders. Arbitrarily shaped polygonal cells are employed to allow more accurate modeling of complex objects without excessive matrix sizes. To demonstrate the validity of the method, the analytical and numerical solutions are compared for triple-layered concentric cylinders. The calculated fields in a model of the human torso exposed to a solenoidal field are also presented. Among other applications, the method should be useful for evaluating electromagnetic (em) biohazards, and for analyzing and designing dielectric antennas and hyperthermia applicators at low frequencies.