  This article presents the convergence analysis of a sequence of piecewise
constant and piecewise linear functions obtained by the Rothe method to the
solution of the first order evolution partial differential inclusion
$u'(t)+Au(t)+\iota^*\partial J(\iota u(t))\ni f(t)$, where the multivalued term
is given by the Clarke subdifferential of a locally Lipschitz functional. The
method provides the proof of existence of solutions alternative to the ones
known in literature and together with any method for underlying elliptic
problem, can serve as the effective tool to approximate the solution
numerically. Presented approach puts into the unified framework known results
for multivalued nonmonotone source term and boundary conditions, and
generalizes them to the case where the multivalued term is defined on the
arbitrary reflexive Banach space as long as appropriate conditions are
satisfied. In addition the results on improved convergence as well as the
numerical examples are presented.
