Full discretisations for nonlinear evolutionary inequalities based on stiffly accurate Runge-Kutta and \(hp\)-finite element methods
type: Item
instance of: scholarly article
title: Full discretisations for nonlinear evolutionary inequalities based on stiffly accurate Runge-Kutta and \(hp\)-finite element methods
zbMATH Open document ID: 1308.65108
author: Yong-Cai Geng
author: Sumit K. Garg
published in: Foundations of Computational Mathematics
publication date: 2015-01-16T00:00:00Z
review text: In this very important work, the authors study the convergence of full discretizations by implicit Runge-Kutta and nonconforming Galerkin methods applied to nonlinear evolutionary inequalities. This work is the first contribution, where a class of implicit Runge-Kutta methods is investigated for nonlinear evolutionary inequalities. The authors justify in more detail the employed novel formulation of the fully discrete variational inequality and deduce the needed auxiliary results for stiffly accurate Runge-Kutta and \(hp\)-finite element discretizations. They introduce a simple though characteristic model problem, which the authors serve as an illustration of the general framework of nonlinear evolutionary inequalities. The authors first state the nonlinear parabolic initial boundary value problem involving the \(r\)-Laplacian and free boundary condition of Signorini type and then indicate its reformulation as evolutionary variational inequality. In order to deduce an appropriate time-discrete variational inequality, the authors initiate the procedure for the time-continous problem. The approximations based on the \(hp\)-finite element method are introduced. The authors introduce a general framework of nonlinear evolutionary variational inequalities involving monotone-convex functionals. Moreover, they deduce an existence and uniqueness result as well as a priori bounds for the discrete solution, which are an essential ingredient in the convergence analysis.
reviewed by: Jan Lovíšek
Mathematics Subject Classification ID: 65K15
Mathematics Subject Classification ID: 49J40
Mathematics Subject Classification ID: 35K61
Mathematics Subject Classification ID: 35K86
Mathematics Subject Classification ID: 47J05
Mathematics Subject Classification ID: 47J20
Mathematics Subject Classification ID: 47J22
Mathematics Subject Classification ID: 47H05
Mathematics Subject Classification ID: 65M12
zbMATH DE Number: 6387174
zbMATH Keywords: nonlinear evolutionary inequalities
zbMATH Keywords: nonlinear differential inclusions
zbMATH Keywords: monotone operators
zbMATH Keywords: stiffly accurate Runge-Kutta methods
zbMATH Keywords: nonconforming Galerkin methods
zbMATH Keywords: \(hp\)-finite element approximations
zbMATH Keywords: stability
zbMATH Keywords: convergence
zbMATH Keywords: Signorini-type initial boundary value problem
zbMATH Keywords: piecewise constant
zbMATH Keywords: time interpolant
zbMATH Keywords: existence theory
zbMATH Keywords: convex subsets
zbMATH Keywords: evolutionary inequalities
zbMATH Keywords: implicit Euler method
zbMATH Keywords: Hilbert space
zbMATH Keywords: lower semicontinuity
zbMATH Keywords: variational inequalities
Wikidata QID: Q40884683
describes a project that uses: RODAS
MaRDI profile type: Publication
full work available at URL: s10208-013-9179-3
OpenAlex ID: W1993750280
cites work: Approximation theory for the \(hp\)-version finite element method and application to the nonlinear Laplacian
cites work: Q4040886
cites work: Q5186236
cites work: Q3324260
cites work: Thepandh-pVersions of the Finite Element Method, Basic Principles and Properties
cites work: Numerical Methods for Ordinary Differential Equations
cites work: A theory of discretization for nonlinear evolution inequalities applied to parabolic Signorini problems
cites work: Q3994914
cites work: Q4273944
cites work: Adaptive \(hp\)-FEM for the contact problem with Tresca friction in linear elasticity: The primal-dual formulation and a posteriori error estimation
cites work: hp-FEM for the Contact Problem with Tresca Friction in Linear Elasticity: The Primal Formulation
cites work: Q4097611
cites work: Q4680559
cites work: Stiffly accurate Runge–Kutta methods for nonlinear evolution problems governed by a monotone operator
cites work: A comparison of the \(h\)-, \(p\)-, \(hp\)-, and \(rp\)-version of the FEM for the solution of the 2D Hertzian contact problem
cites work: Q3321366
cites work: Q3914338
cites work: Finite-element convergence for contact problems in plane linear elastostatics
cites work: Three-field modelling of nonlinear nonsmooth boundary value problems and stability of differential mixed variational inequalities
cites work: \(hp\)-FEM convergence for unilateral contact problems with Tresca friction in plane linear elastostatics
cites work: Q4000061
cites work: Q3995495
cites work: Plasticity. Mathematical theory and numerical analysis
cites work: Solution of variational inequalities in mechanics
cites work: Q3994823
cites work: A \(p\)-version finite element method for nonlinear elliptic variational inequalities in 2D
cites work: Adaptive \(hp\)-versions of boundary element methods for elastic contact problems
cites work: Convergence of convex sets and of solutions of variational inequalities
cites work: On a regularity property and a priori estimates for solutions of nonlinear parabolic variational inequalities
cites work: Q3689524
cites work: Q3821170
cites work: An Introduction to Partial Differential Equations
cites work: Q4220650
cites work: Models and analysis of quasistatic contact. Variational methods
cites work: Analysis and Approximation of Contact Problems with Adhesion or Damage
cites work: Numerik gewöhnlicher Differentialgleichungen
cites work: 1.—Perturbation Theory for Sobolev Spaces

Full discretisations for nonlinear evolutionary inequalities based on stiffly accurate Runge-Kutta and \(hp\)-finite element methods (cites work)
cites work: Q3995594

Full discretisations for nonlinear evolutionary inequalities based on stiffly accurate Runge-Kutta and \(hp\)-finite element methods (DOI)
DOI: 10.1007/S10208-013-9179-3