Second order gauge invariant discretizations to the Schrödinger and Pauli equations

We introduce a numerical method, based on finite elements and lattice gauge theory, to compute approximate solutions to Schrödinger and Pauli equations. The crucial geometric property of the method is discrete gauge invariance. The main new achievement is second order convergence. This is proved by interpreting the method as defined on gauge potential dependent finite element spaces and providing an analysis of such spaces in terms of gauge potential dependent norms on simplices of all dimensions.