Rothe method for parabolic variational-hemivariational inequalities

The paper deals with the convergence analysis of the semidiscrete Rothe scheme for the parabolic variational-hemivariational inequality with the nonlinear pseudomonotone elliptic operator. The problem involves both a discontinuous and nonmonotone multivalued term as well as a monotone term with potentials which assume infinite values and hence are not locally Lipschitz. We prove the existence of a solution and establish a convergence result of a numerical semidiscrete scheme. The proof can be viewed both as the proof of solution existence as well as the proof of the convergence of a numerical semidiscrete scheme. The numerical simulations to present the rate of convergence with respect to space and time for piecewise linear finite elements are presented as well.