  A new interior-exterior penalty method for solving quasi-variational
inequality and pseudo-monotone operators arising in two-dimensional point
contact problem has been analyzed and developed in discontinuous Galerkin
finite volume method environment. In this article, we proved the existence of
solution for the more realistic model problem without taking any constant
assumption in viscosity or density of the lubricant. We have shown that optimal
error estimate of $H^{1}$ and $L^{2}$ norm can be achieved under a light load
non-dimensional parameter condition. In addition, we provided a complete
algorithm to tackle all numerical complexities appear in the solution
procedure. We obtained results for moderate loaded conditions which have been
discussed at the end of the section. Furthermore, results give a hope for the
further development of the scheme for highly loaded condition appeared in a
more realistic operating situation which will be discussed in part II. This
method is well suited for solving elasto-hydrodynamic lubrication line as well
as point contact problems and can probably be treated as commercial software.
