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In this paper, we consider the initial-boundary value problem for thetime-dependent Maxwell--Schr\"{o}dinger equations in the Coulomb gauge. Wefirst prove the global existence of weak solutions to the equations. Next wepropose an energy-conserving fully discrete finite element scheme for thesystem and prove the existence and uniqueness of solutions to the discretesystem. The optimal error estimates for the numerical scheme without anytime-step restrictions are then derived. Numerical results are provided tosupport our theoretical analysis.

Mathematical and numerical analysis of the time-dependent Maxwell--Schr\"{o}dinger Equations in the Coulomb gauge