  We consider the magnetic Dirichlet Laplacian with constant magnetic field on
domains of finite measure. In the case of a disc, we prove that the eigenvalue
branches with respect to the field strength behave asymptotically linear with
an exponentially small remainder term as the field strength goes to infinity.
We compute the asymptotic expression for this remainder term. Furthermore, we
show that for sufficiently large magnetic field strengths, the spectral bound
corresponding to the P\'olya conjecture for the non-magnetic Dirichlet
Laplacian is violated up to a sharp excess factor which is independent of the
domain.
