# Full Discretisations for Nonlinear Evolutionary Inequalities Based on Stiffly Accurate Runge-Kutta and hp-Finite Element Methods

Mechthild Thalhammer*

The convergence of full discretisations by implicit Runge--Kutta and nonconforming Galerkin methods applied to nonlinear evolutionary inequalities is studied. The scope of applications includes differential inclusions governed by a nonlinear operator that is uniformly monotone and fulfills a certain growth condition. A basic assumption on the considered class of stiffly accurate Runge--Kutta time discretisations is a stability criterion which is in particular satisfied by the Radau IIA and Lobatto IIIC methods. In order to allow nonconforming $hp$-finite element approximations of unilateral constraints, set convergence of convex subsets in the sense of Glowinski--Mosco--Stummel is utilised. An appropriate formulation of the fully discrete variational inequality is deduced on the basis of a characteristical application, a $r$-harmonic Signorini initial-boundary value problem. Under hypotheses close to the existence theory of nonlinear first-order evolutionary equations and inequalities involving a monotone main part, a convergence result for the piecewise constant and piecewise linear in time interpolants is established.

Mathematics Subject Classification: 35K86 65M12

Keywords: Nonlinear evolutionary inequalities; Nonlinear differential inclusions; Monotone operators; Stiffly accurate Runge-Kutta methods; Nonconforming Galerkin methods; hp-Finite element approximations; Stability; Convergence

Minisymposion: Nonsmooth and Unilateral Problems - Modelling, Analysis and Optimization Methods