Rothe method for parabolic variational–hemivariational inequalities ☆

Abstract The paper deals with the convergence analysis of the semidiscrete Rothe scheme for the parabolic variational–hemivariational inequality with the nonlinear pseudomonotone elliptic operator. The problem involves both a discontinuous and nonmonotone multivalued term as well as a monotone term with potentials which assume infinite values and hence are not locally Lipschitz. We prove the existence of a solution and establish a convergence result of a numerical semidiscrete scheme. The proof can be viewed both as the proof of solution existence as well as the proof of the convergence of a numerical semidiscrete scheme. The numerical simulations to present the rate of convergence with respect to space and time for piecewise linear finite elements are presented as well.

[1]  Piotr Kalita,et al.  Convergence of Rothe scheme for hemivariational inequalities of parabolic type , 2011, 1108.5469.

[2]  Mikaël Barboteu,et al.  ANALYSIS OF A CONTACT PROBLEM WITH NORMAL COMPLIANCE, FINITE PENETRATION AND NONMONOTONE SLIP DEPENDENT FRICTION , 2014 .

[3]  E. Zeidler.  Nonlinear Functional Analysis and Its Applications: II/ A: Linear Monotone Operators , 1989 .

[4]  H. Nagase.  Addendum to"On an Application of Rothe′s Method to Nonlinear Parabolic Variational Inequalities" , 1989 .

[5]  Nan-Jing Huang,et al.  Existence theorems of the variational-hemivariational inequalities , 2013, J. Glob. Optim..

[6]  Anna Ochal,et al.  Boundary hemivariational inequality of parabolic type , 2004 .

[7]  W. Han,et al.  Discontinuous Galerkin methods for solving the Signorini problem , 2011, IMA Journal of Numerical Analysis.

[8]  Carsten Carstensen,et al.  A theory of discretization for nonlinear evolution inequalities applied to parabolic Signorini problems , 1999 .

[9]  Nikolas S. Papageorgiou,et al.  An introduction to nonlinear analysis , 2003 .

[10]  Stanis lslash,et al.  On existence of solutions for parabolic hemivariational inequalities , 2001 .

[11]  Dumitru Motreanu,et al.  A class of variational–hemivariational inequalities of elliptic type , 2010 .

[12]  P. D. Panagiotopoulos,et al.  On parabolic hemivariational inequalities and applications , 1999 .

[13]  Dumitru Motreanu,et al.  Minimax Theorems and Qualitative Properties of the Solutions of Hemivariational Inequalities , 1998 .

[14]  Cezar Lupu,et al.  On a class of variational-hemivariational inequalities involving set valued mappings , 2009, 0904.4001.

[15]  T. Roubíček.  Nonlinear partial differential equations with applications , 2005 .

[16]  Mikaël Barboteu,et al.  An analytical and numerical approach to a bilateral contact problem with nonmonotone friction , 2013, Int. J. Appl. Math. Comput. Sci..

[17]  Quasilinear parabolic variational inequalities with multi-valued lower-order terms , 2014 .

[18]  P. Panagiotopoulos,et al.  Finite Element Method for Hemivariational Inequalities: Theory, Methods and Applications , 1999 .

[19]  Mircea Sofonea,et al.  Nonlinear Inclusions and Hemivariational Inequalities: Models and Analysis of Contact Problems , 2012 .

[20]  W. Han,et al.  Quasistatic Contact Problems in Viscoelasticity and Viscoplasticity , 2002 .

[21]  Fei Wang,et al.  Discontinuous Galerkin methods for solving a quasistatic contact problem , 2013, Numerische Mathematik.

[22]  V. A. Kovtunenko,et al.  A hemivariational inequality in crack problems , 2011 .

[23]  Directedness of solution set for some quasilinear multi-valued parabolic problems , 2010 .

[24]  Existence and extremal solutions of parabolic variational–hemivariational inequalities , 2013 .

[25]  Minh Ngoc Dao,et al.  Nonconvex bundle method with application to a delamination problem , 2014, Comput. Optim. Appl..