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Semidiscrete variable time-step θ-scheme for nonmonotone evolution inclusion

In this paper we show the convergence of a semidiscrete time stepping θ-scheme on a time grid of variable length to the solution of parabolic operator differential inclusion in the framework of evolution triple. The multifunction is assumed to be strong-weak upper-semicontinuous and to have nonempty, closed and convex values, while the quasilinear operator present in the problem is required to be pseudomonotone, coercive and satisfy the appropriate growth condition. The convergence of piecewise constant and piecewise linear interpolants constructed on the solutions of time discrete problems is shown. Under an additional assumption on the sequence of time grids and regularity of quasilinear operator strong convergence results are obtained.