Optimal error estimates of mixed FEMs for second order hyperbolic integro-differential equations with minimal smoothness on initial data

In this article, mixed finite element methods are discussed for a class of hyperbolic integro-differential equations (HIDEs). Based on a modification of the nonstandard energy formulation of Baker, both semidiscrete and completely discrete implicit schemes for an extended mixed method are analyzed and optimal L∞(L2)L∞(L2)-error estimates are derived under minimal smoothness assumptions on the initial data. Further, quasi-optimal estimates are shown to hold in L∞(L∞)L∞(L∞)-norm. Finally, the analysis is extended to the standard mixed method for HIDEs and optimal error estimates in L∞(L2)L∞(L2)-norm are derived again under minimal smoothness on initial data.