Full Discretisations for Nonlinear Evolutionary Inequalities Based on Stiffly Accurate RungeKutta and hp-Finite Element Methods
The convergence of full discretisations by implicit RungeKutta 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 RungeKutta time discretisations is a stability criterion which is in particular satisfied by the RadauIIA and LobattoIIIC methods. In order to allow nonconforming hp-finite element approximations of unilateral constraints, set convergence of convex subsets in the sense of GlowinskiMoscoStummel 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.
nonlinear evolutionary inequalities convergence nonconforming galerkin methods stability hp-finite element approximations nonlinear differential inclusions monotone operators stiffly accurate rungekutta methods 35k61 35k86 47j05 47j20 47j22 47h05 65m12