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We introduce a numerical method, based on finite elements and lattice gaugetheory, to compute approximate solutions to Schr\"odinger and Pauli equations.The crucial geometric property of the method is discrete gauge invariance. Themain new achievement is second order convergence. This is proved byinterpreting the method as defined on gauge potential dependent finite elementspaces and providing an analysis of such spaces in terms of gauge potentialdependent norms on simplices of all dimensions.

Second order gauge invariant discretizations to the Schr\"odinger and Pauli equations