title: Full Discretisations for Nonlinear Evolutionary Inequalities Based on Stiffly Accurate Runge-Kutta andhp-Finite Element Methods.
abstract: 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 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 nonconforminghp-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 characteristic example of use, a Signorini-type 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 in time interpolant is established.
authors.affiliation: Institut für Mathematik und Rechneranwendung, Universität der Bundeswehr München, Werner-Heisenberg-Weg 39, 85577 Neubiberg, Germany.
authors.forename: J
authors.initials: J
authors.lastname: Gwinner
authors.affiliation: Institut für Mathematik, Leopold-Franzens Universität Innsbruck, Technikerstrasse 13/7, 6020 Innsbruck, Austria.
authors.forename: M
authors.initials: M
authors.lastname: Thalhammer
keywords: Convergence; Monotone operators; Nonconforming Galerkin methods; Nonlinear differential inclusions; Nonlinear evolutionary inequalities; Stability; Stiffly accurate Runge–Kutta methods; hp-finite element approximations