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A novel finite-element formulation for the analysis of the energy levels of a two-dimensional quantum cavity in a magnetic field is proposed. Our formulation is based on both a variational and Galerkin methods which, when combined with a finite-element method, lead to eigenvalue problems characterized by either real or Hermitian matrices contrary to algorithms proposed in the past. These eigenvalue problems can then be solved quite efficiently on a computer using standard library packages. Also, it is confirmed that the results obtained here are independent of the choice of gauge.