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Adaptive algorithms for automatic finite element mesh generation as used for two and three-dimensional analysis of electromagnetic devices require global and local refinement criteria to indicate when and where to refine the mesh. As the refinement criteria, local and global estimates of the errors in a calculated solution are proposed using standard and complementary finite element techniques based on Synges hypercircle method. Two local refinement criteria are presented which bound element by element errors between the exact and calculated energy density and stored energy of each element. Meshes generated using these criteria are shown to be well refined in areas of concern for finite element analysis. Illustrative examples demonstrate these criteria for some diverse applications.