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

• Published in: Foundations of computational mathematics Vol. 14; no. 5; pp. 913 - 949
• Main Authors: Gwinner, J., Thalhammer, M.
• Format: Journal Article
• Language: English
• Published: Boston Springer US 01.10.2014; Springer; Springer Nature B.V
• Subjects: Analysis; Applications of Mathematics; Approximation theory; Chemical elements; Computer Science; Constants; Convergence; Economics; Evolutionary; Finite element method; Galerkin methods; Inequalities; Inequalities (Mathematics); Linear and Multilinear Algebras; Math Applications in Computer Science; Mathematical analysis; Mathematics; Mathematics and Statistics; Matrix Theory; Nonlinear equations; Nonlinearity; Numerical Analysis; Runge-Kutta method; United States; 35K86; finite element approximations; Stability; 35K61; Nonlinear evolutionary inequalities; Monotone operators; Stiffly accurate Runge–Kutta methods; Nonlinear differential inclusions; Convergence; 47J22; 47J20; 47J05; 47H05; Nonconforming Galerkin methods; 65M12; hp-finite element approximations
• ISSN: 1615-3375; 1615-3383
• DOI: 10.1007/s10208-013-9179-3

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.