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We propose a method to compute approximate eigenpairs of the Schrödinger operator on a bounded domain in the presence of an electromagnetic field. Formulated for simplicial meshes, the method combines techniques from lattice gauge theory and finite elements, retaining the discrete gauge invariance of the former but allowing for noncongruent space elements as in the latter. The error of the method is studied in the framework of Strang's variational crimes, comparing with a standard Galerkin approach. When the meshes are quasi-uniform and satisfy the discrete maximum principle, we prove that for a smooth electromagnetic field the crime is of the order of the mesh width $h$, for a Coulomb potential it is of order $h|\log h|$, and for a general finite energy electromagnetic field it is of order $h^{1/2}$.

A Gauge Invariant Discretization on Simplicial Grids of the Schrödinger Eigenvalue Problem in an Electromagnetic Field