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A new interior-exterior penalty method for solving quasi-variationalinequality and pseudo-monotone operators arising in two-dimensional pointcontact problem has been analyzed and developed in discontinuous Galerkinfinite volume method environment. In this article, we proved the existence ofsolution for the more realistic model problem without taking any constantassumption in viscosity or density of the lubricant. We have shown that optimalerror estimate of $H^{1}$ and $L^{2}$ norm can be achieved under a light loadnon-dimensional parameter condition. In addition, we provided a completealgorithm to tackle all numerical complexities appear in the solutionprocedure. We obtained results for moderate loaded conditions which have beendiscussed at the end of the section. Furthermore, results give a hope for thefurther development of the scheme for highly loaded condition appeared in amore realistic operating situation which will be discussed in part II. Thismethod is well suited for solving elasto-hydrodynamic lubrication line as wellas point contact problems and can probably be treated as commercial software.

Interior-exterior penalty approach for solving elasto-hydrodynamic lubrication problem: Part I