Rothe method for parabolic variational–hemivariational inequalities

Abstract

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.