paper, we introduce DEC-QED, a computational approach for modeling the electrodynamics of superconducting electronic circuits containing Josephson junctions in arbitrary three-dimensional electromagnetic environments. DEC-QED captures the non-linear response and induced currents of BCS superconductors and accurately captures phenomena such as the Meissner effect, flux quantization and Josephson effects. Using a finite-element construction based on Discrete Exterior Calculus (DEC), DEC-QED can accurately simulate transient and long-time dynamics in superconductors. The expression of the entire electrodynamic problem in terms of the gauge-invariant flux field and charges makes the resulting classical field theory suitable for second quantization.