  We investigate the type of dual superconductivity responsible for quark
confinement. For this purpose, we solve the field equations of the $U(1)$
Abelian--Higgs model to obtain the static vortex solution in the whole range
without restricting to the long-distance region. Then we use the resulting
magnetic field of the vortex to fit the gauge-invariant chromoelectric field
connecting a pair of quark and antiquark which was measured by numerical
simulations for $SU(2)$ and $SU(3)$ Yang--Mills theories on a lattice. This
result improves the accuracy of the fitted value for the Ginzburg--Landau
parameter to reconfirm the type I dual superconductivity for quark confinement,
which was claimed by preceding works based on an approximate method based on
the Clem ansatz. Moreover, we calculate the Maxwell stress tensor for the
fitted model to obtain the distribution of the force around the flux tube. This
suggests that the attractive force acts on the surface perpendicular to the
chromoelectric flux tube, in agreement with the type I dual superconductivity.
