Search

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 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 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. Keywords Nonlinear evolutionary inequalities * Nonlinear differential inclusions * Monotone operators * Stiffly accurate Runge-Kutta methods * Nonconforming Galerkin methods * hp-finite element approximations * Stability * Convergence Mathematics Subject Classification (2010) 35K61 * 35K86 * 47J05 * 47J20 * 47J22 * 47H05 * 65M12, 1 Introduction Scope of Applications Nowadays, transient heat transfer between different materials under nonlinear unilateral/bilateral transmission conditions [13] and time-dependent contact between deformable bodies with possibly nonlinear material behaviour [14, [...]