  An extensive study of the compact $U(1)$ lattice gauge theory with a higher
derivative gauge-fixing term and a suitable counter-term has been undertaken to
determine the nature of the possible continuum limits for a wide range of the
parameters, especially at strong gauge couplings ($g>1$), adding to our
previous study at a single gauge coupling $g=1.3$ \cite{DeSarkar2016}. Our
major conclusion is that a continuum limit of free massless photons (with the
redundant pure gauge degrees of freedom decoupled) is achieved at any gauge
coupling, not necessarily small, provided the coefficient $\tilde{\kappa}$ of
the gauge-fixing term is sufficiently large. In fact, the region of continuous
phase transition leading to the above physics in the strong gauge coupling
region is found to be analytically connected to the point $g=0$ and
$\tilde{\kappa} \rightarrow \infty$ where the classical action has a global
unique minimum, around which weak coupling perturbation theory in bare
parameters is defined, controlling the physics of the whole region. A second
major conclusion is that, local algorithms like Multihit Metropolis fail to
produce faithful field configurations with large values of the coefficient
$\tilde{\kappa}$ of the higher derivative gauge-fixing term and at large
lattice volumes. A global algorithm like Hybrid Monte Carlo, although at times
slow to move out of metastabilities, generally is able to produce faithful
configurations and has been used extensively in the current study.
