Finite element methods for parabolic variational inequalities with a Volterra term

In this article, we study parabolic integro-differential equations with an obstacle which gives rise to parabolic variational inequalities with a Volterra term. By the introduction of a suitable penalty operator, the given variational inequality is transformed into a variational equality formulation. Then, existence, uniqueness, and regularity results are derived using a priori bounds and compactness arguments. For numerical approximations, finite element Galerkin methods are applied to the penalized problem and error estimates in the energy norm are established for the semidiscrete case. Finally, a backward Euler method combined with rectangle quadrature rule for the integral term is used for the temporal discretization and by coupling the-penalty parameter epsilon, the spatial discretization parameter h and time step size k, a priori error bounds are obtained in suitable norms.