An L2-stability analysis of a θ-scheme for a class of nonlinear parabolic variational inequalities of obstacle type

This paper analyzes the stability of a fully discrete finite element approximation for a class of nonlinear parabolic variational inequalities of obstacle type. The temporal discretization is based on a θ-scheme. We derive a stability condition for the scheme that depends critically on the parameter θ. We prove that the method is unconditionally stable in the L2-norm for θ in 1/2,1. For θ in [0,1/2), we establish a precise Courant-Friedrichs-Lewy (CFL)-type condition, Delta t<=2gγ/(L2(1-2θ)), where γ is the coercivity constant and L is the Lipschitz constant of the nonlinear source term. The analysis is based on a careful choice of test functions in the variational inequality and by deriving sharp estimates of the associated bilinear form.