Variational-Hemivariational Inequalities: A Brief Survey on Mathematical Theory and Numerical Analysis

Weimin Han

doi:10.65135/atsf.2026.6

Authors

Weimin Han Author

DOI:

https://doi.org/10.65135/atsf.2026.6

Keywords:

Variational inequality, hemivariational inequality, variational-hemivariational inequality, well-posedness, numerical solution, finite element method, discontinuous Galerkin method, virtual element method, convergence, error estimate, contact mechanics, Stokes hemivariational inequality, Navier-Stokes hemivariational inequality

Abstract

Variational-hemivariational inequalities are an area full of interesting and challenging mathematical problems. The area can be viewed as a natural extension of that of variational inequalities. Variational-hemivariational inequalities are valuable for application problems from physical sciences and engineering that involve non-smooth and even set-valued relations, monotone or non-monotone, among physical quantities. In the recent years, there has been substantial growth of research interest in modeling, well-posedness analysis, development of numerical methods and numerical algorithms of variational-hemivariational inequalities. This survey paper is devoted to a brief account of well-posedness and numerical analysis results for variational-hemivariational inequalities. The theoretical results are presented for a family of abstract stationary variational-hemivariational inequalities and the main idea is explained for an accessible proof of existence and uniqueness. To better appreciate the distinguished feature of variational-hemivariational inequalities, for comparison, three mechanical problems are introduced leading to a variational equation, a variational inequality, and a variational-hemivariational inequality, respectively. The paper also comments on mixed variational-hemivariational inequalities, with examples from applications in fluid mechanics, and on results concerning the numerical solution of other types (nonstationary, history dependent) of variational-hemivariational inequalities.

References

R. A. Adams and J. J. F. Fournier, Sobolev Spaces. Second edition, Academic Press, New York, 2003.

F. Aldakheel, B. Hudobivnik, E. Artioli, L. Beirao da Veiga and P. Wriggers, Curvilinear virtual elements for contact mechanics. Comput. Methods Appl. Mech. Engrg. 372 (2020), paper no. 113394. DOI: https://doi.org/10.1016/j.cma.2020.113394 DOI: https://doi.org/10.1016/j.cma.2020.113394

W. Akram and M. T. Mohan, Mixed finite element method for a hemivariational inequality of s stationary convective Brinkman-Forchheimer extended Darcy equations. arXiv: 2508.02797.

S. S. Antman, The influence of elasticity in analysis: modern developments. Bulletin of the American Mathematical Society 9 (1983), no. 3, 267--291. DOI: https://doi.org/10.1090/S0273-0979-1983-15185-6

K. Atkinson and W. Han, Theoretical Numerical Analysis: A Functional Analysis Framework. Third edition, Springer, New York, 2009.

M. Barboteu, K. Bartosz, and W. Han, Numerical analysis of an evolutionary variational-hemivariational inequality with application in contact mechanics. Comput. Methods Appl. Mech. Engrg. 318 (2017), 882--897. DOI: https://doi.org/10.1016/j.cma.2017.02.003 DOI: https://doi.org/10.1016/j.cma.2017.02.003

M. Barboteu, K. Bartosz, W. Han and T. Janiczko, Numerical analysis of a hyperbolic hemivariational inequality arising in dynamic contact. SIAM Journal on Numerical Analysis 53 (2015), 527--550. DOI: https://doi.org/10.1137/140969737 DOI: https://doi.org/10.1137/140969737

M. Barboteu, W. Han and S. Migorski, On numerical approximation of a variational-hemivariational inequality modeling contact problems for locking materials. Comput. Math. Appl. 77 (2019), 2894--2905. DOI: https://doi.org/10.1016/j.camwa.2018.08.004 DOI: https://doi.org/10.1016/j.camwa.2018.08.004

K. Bartosz, Numerical analysis of a nonmonotone dynamic contact problem of a non-clamped piezoelectric viscoelastic body. Evol. Equ. Control Theory 9 (2020), 961--980. DOI: https://doi.org/10.3934/eect.2020059 DOI: https://doi.org/10.3934/eect.2020059

L. Beirao da Veiga, F. Brezzi, A. Cangiani, G. Manzini, L. D. Marini and A. Russo, Basic principles of virtual element methods. Math. Models Methods Appl. Sci. 23 (2013), 119--214. DOI: https://doi.org/10.1142/S0218202512500492 DOI: https://doi.org/10.1142/S0218202512500492

L. Beirao da Veiga, F. Brezzi and L. D. Marini, Virtual elements for linear elasticity problems. SIAM J. Numer. Anal. 51 (2013), 794--812. DOI: https://doi.org/10.1137/120874746 DOI: https://doi.org/10.1137/120874746

S. C. Brenner and L. R. Scott, The Mathematical Theory of Finite Element Methods. Third edition, Springer-Verlag, New York, 2008. DOI: https://doi.org/10.1007/978-0-387-75934-0

H. Brezis, Problemes unilateraux, J. Math. Pures et Appl. 51 (1972), 1--168. DOI: https://doi.org/10.1007/BF02764814

H. Brezis, Functional Analysis, Sobolev Spaces and Partial Differential Equations. Springer, New York, 2011. DOI: https://doi.org/10.1007/978-0-387-70914-7

S. Carl and V. K. Le, Multi-Valued Variational Inequalities and Inclusions. Springer, New York, 2021. DOI: https://doi.org/10.1007/978-3-030-65165-7

S. Carl, V. K. Le, and D. Motreanu, Nonsmooth Variational Problems and Their Inequalities: Comparison Principles and Applications. Springer, New York, 2007. DOI: https://doi.org/10.1007/978-0-387-46252-3

G. Caselli, M. Hensel and I. Yousept, Quasilinear variational inequalities in ferromagnetic shielding: well-posedness, regularity, and optimal control. SIAM J. Control Optim. 61 (2023), 2043--2068. DOI: https://doi.org/10.1137/22M1530483 DOI: https://doi.org/10.1137/22M1530483

J. Cen, S. Migorski, C. Min and J.-C. Yao, Hemivariational inequality for contaminant reaction–diffusion model of recovered fracturing fluid in the wellbore of shale gas reservoir. Commun. Nonlinear Sci. Numer. Simul. 118 (2023), paper no. 107020. DOI: https://doi.org/10.1016/j.cnsns.2022.107020 DOI: https://doi.org/10.1016/j.cnsns.2022.107020

X.-L. Cheng, X. Shen, X. Wang and K. Liang, A deep neural network-based method for solving obstacle problems. Nonlinear Anal. Real World Appl. 72 (2023), paper no. 103864. DOI: https://doi.org/10.1016/j.nonrwa.2023.103864 DOI: https://doi.org/10.1016/j.nonrwa.2023.103864

F. Chouly, P. Hild and Y. Renard, Finite Element Approximation of Contact and Friction in Elasticity. Birkhauser/Springer, Cham, 2023. DOI: https://doi.org/10.1007/978-3-031-31423-0

P. G. Ciarlet, The Finite Element Method for Elliptic Problems. North Holland, Amsterdam, 1978. DOI: https://doi.org/10.1115/1.3424474

M. Cihan, B. Hudobivnik, J. Korelc and P. Wriggers, A virtual element method for 3D contact problems with non-conforming meshes. Comput. Methods Appl. Mech. Engrg. 402 (2022), paper no. 115385. DOI: https://doi.org/10.1016/j.cma.2022.115385 DOI: https://doi.org/10.1016/j.cma.2022.115385

F. H. Clarke, Generalized gradients and applications. Trans. Am. Math. Soc. 205 (1975), 247--262. DOI: https://doi.org/10.1090/S0002-9947-1975-0367131-6

F. H. Clarke, Optimization and Nonsmooth Analysis. Wiley, New York, 1983.

F. H. Clarke, Functional Analysis, Calculus of Variations and Optimal Control. Springer, London, 2013. DOI: https://doi.org/10.1007/978-1-4471-4820-3

F. H. Clarke, Y. S. Ledyaev, R. J. Stern and P. R. Wolenski, Nonsmooth Analysis and Control Theory. Springer, New York, 1998.

Z. Denkowski, S. Migorski and N. S. Papageorgiou, An Introduction to Nonlinear Analysis: Applications. Kluwer Academic/Plenum Publishers, Boston, Dordrecht, London, New York, 2003. DOI: https://doi.org/10.1007/978-1-4419-9156-0

G. Duvaut and J.-L. Lions, Inequalities in Mechanics and Physics. Springer-Verlag, Berlin, 1976. DOI: https://doi.org/10.1007/978-3-642-66165-5

C. Eck, J. Jaruvsek, and M. Krbec, Unilateral Contact Problems: Variational Methods and Existence Theorems. Pure and Applied Mathematics, vol. 270, Chapman/CRC Press, New York, 2005.

I. Ekeland and R. Temam, Convex Analysis and Variational Problems, North-Holland, Amsterdam, 1976.

L. C. Evans, Partial Differential Equations. Second edition, American Mathematical Society, 2010.

R. S. Falk, Error estimates for the approximation of a class of variational inequalities. Mathematics of Computation 28 (1974), 963--971. DOI: https://doi.org/10.1090/S0025-5718-1974-0391502-8

C. Fang, K. Czuprynski, W. Han, X. L. Cheng and X. Dai, Finite element method for a stationary Stokes hemivariational inequality with slip boundary condition. IMA J. Numer. Anal. 40 (2020), 2696--2716. DOI: https://doi.org/10.1093/imanum/drz032 DOI: https://doi.org/10.1093/imanum/drz032

F. Feng, W. Han and J. Huang, Virtual element method for elliptic hemivariational inequalities. Journal of Scientific Computing 81 (2019), 2388--2412. DOI: https://doi.org/10.1016/j.cam.2020.113330 DOI: https://doi.org/10.1007/s10915-019-01090-2

F. Feng, W. Han and J. Huang, Virtual element method for elliptic hemivariational inequalities with a convex constraint. Numerical Mathematics: Theory, Methods and Applications 14 (2021), 589--612. DOI: https://doi.org/10.4208/nmtma.OA-2020-0180 DOI: https://doi.org/10.4208/nmtma.OA-2020-0180

F. Feng, W. Han and J. Huang, The virtual element method for an obstacle problem of a Kirchhoff plate. Commun. Nonlinear Sci. Numer. Simul. 103 (2021), paper no. 106008. DOI: https://doi.org/10.1016/j.cnsns.2021.106008 DOI: https://doi.org/10.1016/j.cnsns.2021.106008

F. Feng, W. Han and J. Huang, A nonconforming virtual element method for a fourth-order hemivariational inequality in Kirchhoff plate problem. Journal of Scientific Computing 90 (2022), paper no. 89. DOI: https://doi.org/10.1007/s10915-022-01759-1 DOI: https://doi.org/10.1007/s10915-022-01759-1

G. Fichera, Problemi elastostatici con vincoli unilaterali: il problema di Signorini con ambigue condizioni al contorno. Atti Accad. Naz. Lincei, Mem. Cl. Sci. Fis. Mat. Nat., Sez. I, VIII. Ser. 7 (1964), 91--140.

H. Fujita, Flow Problems with Unilateral Boundary Conditions, College de France. Lecons, 1993.

H. Fujita, A mathematical analysis of motions of viscous incompressible fluid under leak or slip boundary conditions. RIMS Kokyuroku 888 (1994), 199--216.

R. Glowinski, Numerical Methods for Nonlinear Variational Problems, Springer-Verlag, New York, 1984. DOI: https://doi.org/10.1007/978-3-662-12613-4

R. Glowinski, J.-L. Lions, and R. Tr'{e}moli`{e}res, Numerical Analysis of Variational Inequalities, North-Holland, Amsterdam, 1981.

D. Goeleven and D. Motreanu, Variational and Hemivariational Inequalities: Theory, Methods and Applications. Vol. I. Unilateral Analysis and Unilateral Mechanics, Kluwer Academic Publishers, Boston, MA, 2003. DOI: https://doi.org/10.1007/978-1-4419-8610-8_2

D. Goeleven, D. Motreanu, Y. Dumont and M. Rochdi, Variational and Hemivariational Inequalities: Theory, Methods and Applications. Vol. II. Unilateral Problems, Kluwer Academic Publishers, Boston, MA, 2003. DOI: https://doi.org/10.1007/978-1-4419-8610-8

J. Gwinner, B. Jadamba, A. A. Khan, and F. Raciti, Uncertainty Quantification in Variational Inequalities: Theory, Numerics, and Applications. CRC Press, Boca Raton, Florida, 2022. DOI: https://doi.org/10.1201/9781315228969

D. Han and W. Han, Numerical analysis of an evolutionary variational-hemivariational inequality with application to a dynamic contact problem. Journal of Computational and Applied Mathematics 358 (2019), 163--178. DOI: https://doi.org/10.1016/j.cam.2019.03.010 DOI: https://doi.org/10.1016/j.cam.2019.03.010

W. Han, Numerical analysis of stationary variational-hemivariational inequalities with applications in contact mechanics. Mathematics and Mechanics of Solids 23 (2018), 279--293. DOI: https://doi.org/10.1177/1081286517713342 DOI: https://doi.org/10.1177/1081286517713342

W. Han, Minimization principles for elliptic hemivariational inequalities. Nonlinear Analysis: Real World Applications 54 (2020), paper no. 103114. DOI: https://doi.org/10.1016/j.nonrwa.2020.103114 DOI: https://doi.org/10.1016/j.nonrwa.2020.103114

W. Han, A revisit of elliptic variational-hemivariational inequalities. Numerical Functional Analysis and Optimization 42 (2021), 371--395. DOI: https://doi.org/10.1080/01630563.2021.1881541 DOI: https://doi.org/10.1080/01630563.2021.1881541

W. Han, An Introduction to Theory and Applications of Stationary Variational-Hemivariational Inequalities. Springer, New York, 2024. DOI: https://doi.org/10.1007/978-3-031-74216-3

W. Han, K. Czuprynski and F. Jing, Mixed finite element method for a hemivariational inequality of stationary Navier-Stokes equations. Journal of Scientific Computing 89 (2021), paper no. 8. DOI: https://doi.org/10.1007/s10915-021-01614-9 DOI: https://doi.org/10.1007/s10915-021-01614-9

W. Han, F. Feng, F. Wang and J. Huang, Numerical analysis of hemivariational inequalities with applications in contact mechanics. Advances in Applied Mechanics 60 (2025), 113--178. DOI: https://doi.org/10.1016/bs.aams.2025.02.003

W. Han, F. Jing and Y. Yao, Stabilized mixed finite element methods for a Navier--Stokes hemivariational inequality. BIT Numerical Mathematics 63 (2023), paper no. 46. DOI: https://doi.org/10.1007/s10543-023-00985-9 DOI: https://doi.org/10.1007/s10543-023-00985-9

W. Han, M. Ling and F. Wang, Numerical solution of an H(curl)-elliptic hemivariational inequality. IMA J. Numer. Anal. 43 (2023), 976--1000. DOI: https://doi.org/10.1093/imanum/drac007 DOI: https://doi.org/10.1093/imanum/drac007

W. Han and A. Matei, Minimax principles for elliptic mixed hemivariational-variational inequalities. Nonlinear Analysis: Real World Applications 64 (2022), paper no. 103448. DOI: https://doi.org/10.1016/j.nonrwa.2021.103448 DOI: https://doi.org/10.1016/j.nonrwa.2021.103448

W. Han and A. Matei, Well-posedness of a general class of elliptic mixed hemivariational-variational inequalities. Nonlinear Analysis: Real World Applications 66 (2022), 103553. DOI: https://doi.org/10.1016/j.nonrwa.2022.103553 DOI: https://doi.org/10.1016/j.nonrwa.2022.103553

W. Han, S. Migorski and M. Sofonea, A class of variational-hemivariational inequalities with applications to frictional contact problems. SIAM Journal on Mathematical Analysis 46 (2014), 3891--3912. DOI: https://doi.org/10.1137/140963248 DOI: https://doi.org/10.1137/140963248

W. Han and M. Nashed, On variational-hemivariational inequalities in Banach spaces. Communications in Nonlinear Science and Numerical Simulation 124 (2023), paper no. 107309. DOI: https://doi.org/10.1016/j.cnsns.2023.107309 DOI: https://doi.org/10.1016/j.cnsns.2023.107309

W. Han, H. Qiu and L. Mei, On a Stokes hemivariational inequality for incompressible fluid flows with damping. Nonlinear Analysis: Real World Applications 79 (2024), paper no. 104131. DOI: https://doi.org/10.1016/j.nonrwa.2024.104131 DOI: https://doi.org/10.1016/j.nonrwa.2024.104131

W. Han and B. D. Reddy, Plasticity: Mathematical Theory and Numerical Analysis. Second edition, Springer-Verlag, 2013. DOI: https://doi.org/10.1007/978-1-4614-5940-8

W. Han and M. Sofonea, Quasistatic Contact Problems in Viscoelasticity and Viscoplasticity. American Mathematical Society and International Press, 2002. DOI: https://doi.org/10.1090/amsip/030

W. Han and M. Sofonea, Numerical analysis of hemivariational inequalities in contact mechanics. Acta Numerica 28 (2019), 175--286. DOI: https://doi.org/10.1017/S0962492919000023 DOI: https://doi.org/10.1017/S0962492919000023

W. Han, M. Sofonea and M. Barboteu, Numerical analysis of elliptic hemivariational inequalities. SIAM J. Numer. Anal. 55 (2017), 640--663. DOI: https://doi.org/10.1137/16M1072085 DOI: https://doi.org/10.1137/16M1072085

W. Han, M. Sofonea and D. Danan, Numerical analysis of stationary variational-hemivariational inequalities. Numer. Math. 139 (2018), 563--592. DOI: https://doi.org/10.1007/s00211-018-0951-9 DOI: https://doi.org/10.1007/s00211-018-0951-9

W. Han and S. Zeng, On convergence of numerical methods for variational-hemivariational inequalities under minimal solution regularity. Applied Mathematics Letters 93 (2019), 105--110. DOI: https://doi.org/10.1016/j.aml.2019.02.007 DOI: https://doi.org/10.1016/j.aml.2019.02.007

P. Hartman and G. Stampacchia, On some nonlinear elliptic differential functional equations. Acta Math. 15 (1966), 271--310. DOI: https://doi.org/10.1007/BF02392210

J. Haslinger, I. Hlavacek and J. Necas, Numerical methods for unilateral problems in solid mechanics. In Handbook of Numerical Analysis, Vol. IV, P.G. Ciarlet and J.L. Lions, eds., North-Holland, Amsterdam, 1996, 313--485. DOI: https://doi.org/10.1016/S1570-8659(96)80005-6

J. Haslinger, M. Miettinen and P. D. Panagiotopoulos, Finite Element Method for Hemivariational Inequalities: Theory, Methods and Applications. Kluwer Academic Publishers, Boston, Dordrecht, London, 1999. DOI: https://doi.org/10.1007/978-1-4757-5233-5

I. Hlavacek, J. Haslinger, J. Necas and J. Lovisek, Solution of Variational Inequalities in Mechanics. Springer-Verlag, New York, 1988. DOI: https://doi.org/10.1007/978-1-4612-1048-1

J. Huang, C. Wang and H. Wang, A deep learning method for elliptic hemivariational inequalities. East Asian J. Appl. Math. 12 (2022), 487--502. DOI: https://doi.org/10.4208/eajam.081121.161121%20 DOI: https://doi.org/10.4208/eajam.081121.161121

A. Jayswal and T. Antczak (eds.), Continuous Optimization and Variational Inequalities. CRC Press, Boca Raton, 2023. DOI: https://doi.org/10.1201/9781003289883

M. Jureczka and A. Ochal, A nonsmooth optimization approach for hemivariational inequalities with applications to contact mechanics. Appl. Math. Optim. 83 (2021), 1465--1485. DOI: https://doi.org/10.1007/s00245-019-09593-y DOI: https://doi.org/10.1007/s00245-019-09593-y

M. Jureczka, A. Ochal and P. Bartman, A nonsmooth optimization approach for time-dependent hemivariational inequalities. Nonlinear Anal. Real World Appl. 73 (2023), paper no. 103871. DOI: https://doi.org/10.1016/j.nonrwa.2023.103871 DOI: https://doi.org/10.1016/j.nonrwa.2023.103871

N. Kikuchi and J. T. Oden, Contact Problems in Elasticity: A Study of Variational Inequalities and Finite Element Methods. SIAM, Philadelphia, 1988. DOI: https://doi.org/10.1137/1.9781611970845

M. Ling, W. Han and S. Zeng, A pressure projection stabilized mixed finite element method for a Stokes hemivariational inequality. Journal of Scientific Computing 92 (2022), paper no. 13. DOI: https://doi.org/10.1007/s10915-022-01871-2}{DOI: 10.1007/s10915-022-01871-2} DOI: https://doi.org/10.1007/s10915-022-01871-2

M. Ling, F. Wang and W. Han, The nonconforming virtual element method for a stationary Stokes hemivariational inequality with slip boundary condition. Journal of Scientific Computing 85 (2020), paper no. 56. DOI: https://doi.org/10.1007/s10915-020-01333-7 DOI: https://doi.org/10.1007/s10915-020-01333-7

M. Ling, W. Xiao and W. Han, Numerical analysis of a history-dependent mixed hemivariational-variational inequality in contact problems. Comput. Math. Appl. 166 (2024), 65--76. DOI: https://doi.org/10.1016/j.camwa.2024.04.024 DOI: https://doi.org/10.1016/j.camwa.2024.04.024

J.-L. Lions and G. Stampacchia, Variational inequalities. Comm. Pure Appl. Math. 20 (1967), 493--519. DOI: https://doi.org/10.1002/cpa.3160200302

S. Migorski, Y. Chao, J. He and S. Dudek, Analysis of quasi-variational–hemivariational inequalities with applications to Bingham-type fluids. Commun. Nonlinear Sci. Numer. Simul. 133 (2024), paper no. 107968. DOI: https://doi.org/10.1016/j.cnsns.2024.107968 DOI: https://doi.org/10.1016/j.cnsns.2024.107968

S. Migorski and S. Dudek, A class of variational–hemivariational inequalities for Bingham type fluids. Applied Mathematics & Optimization 85, paper no. 16. DOI: https://doi.org/10.1007/s00245-022-09855-2}{DOI: 10.1007/s00245-022-09855-2 DOI: https://doi.org/10.1007/s00245-022-09855-2

S. Migorski, A. Ochal and M. Sofonea, Nonlinear Inclusions and Hemivariational Inequalities. Models and Analysis of Contact Problems. Advances in Mechanics and Mathematics 26, Springer, New York, 2013. DOI: https://doi.org/10.1007/978-1-4614-4232-5

D. Motreanu and P. D. Panagiotopoulos, Minimax Theorems and Qualitative Properties of the Solutions of Hemivariational Inequalities. Kluwer Academic Publishers, Berlin, 1999. DOI: https://doi.org/10.1007/978-1-4615-4064-9

Z. Naniewicz and P. D. Panagiotopoulos, Mathematical Theory of Hemivariational Inequalities and Applications. Dekker, New York, 1995.

J. Nevcas and I. Hlavavcek, Mathematical Theory of Elastic and Elastoplastic Bodies: An Introduction. Elsevier, Amsterdam, 1981.

A. Ochal, M. Jureczka and P. Bartman, A survey of numerical methods for hemivariational inequalities with applications to contact mechanics. Commun. Nonlinear Sci. Numer. Simul. 114 (2022), paper no. 106563. DOI: https://doi.org/10.1016/j.cnsns.2022.106563 DOI: https://doi.org/10.1016/j.cnsns.2022.106563

P. D. Panagiotopoulos, Nonconvex energy functions, hemivariational inequalities and substationary principles. Acta Mechanica 42 (1983), 160--183.

P. D. Panagiotopoulos, Hemivariational Inequalities, Applications in Mechanics and Engineering. Springer-Verlag, Berlin, 1993. DOI: https://doi.org/10.1007/978-3-642-51677-1

J. Qiu, M. Ling, F. Wang and B. Wu, Nonconforming finite element method for a 4th-order history-dependent hemivariational inequality. Commun. Nonlinear Sci. Numer. Simul. 145 (2025), paper no. 108750. DOI: https://doi.org/10.1016/j.cnsns.2025.108750 DOI: https://doi.org/10.1016/j.cnsns.2025.108750

J. Qiu, F. Wang, M. Ling and J. Zhao, The interior penalty virtual element method for the fourth-order elliptic hemivariational inequality. Commun. Nonlinear Sci. Numer. Simul. 127 (2023), paper no. 107547. DOI: https://doi.org/10.1016/j.cnsns.2023.107547}{DOI: 10.1016/j.cnsns.2023.107547 DOI: https://doi.org/10.1016/j.cnsns.2023.107547

A. Signorini, Sopra alcune questioni di elastostatica. Atti della Societ`a Italiana per il Progresso delle Scienze, 1933.

M. Sofonea, W. Han, and M. Shillor, Analysis and Approximation of Contact Problems with Adhesion or Damage. Chapman & Hall/CRC, New York, 2006. DOI: https://doi.org/10.1201/9781420034837

M. Sofonea and A. Matei, Variational Inequalities with Applications: A Study of Antiplane Frictional Contact Problems. Springer, 2009. DOI: https://doi.org/10.1007/978-0-387-87460-9_5

M. Sofonea and A. Matei, History-dependent quasivariational inequalities arising in contact mechanics. European Journal of Applied Mathematics 22 (2011), 471--491. DOI: https://doi.org/10.1017/S0956792511000192

M. Sofonea and A. Matei, Mathematical Models in Contact Mechanics. Cambridge University Press, Cambridge, 2012. DOI: https://doi.org/10.1017/CBO9781139104166

M. Sofonea and S. Migorski, A class of history-dependent variational-hemivariational inequalities. Nonlinear Differ. Equ. Appl. 23 (2016), paper no. 38. DOI: https://doi.org/10.1007/s00030-016-0391-0 DOI: https://doi.org/10.1007/s00030-016-0391-0

M. Sofonea and S. Migorski, Variational-Hemivariational Inequalities with Applications. Second edition, CRC Press, Boca Raton, FL, 2025. DOI: https://doi.org/10.1201/9781003452782

G. Stampacchia, Formes bilineaires coercitives sur les ensembles convexes. C. R. Acad. Sci. 258 (1964), 4413--4416.

X. Sun, X.-L. Cheng and H. Xuan, Error estimates and numerical simulations of a thermoviscoelastic contact problem with damage and long memory. Commun. Nonlinear Sci. Numer. Simul. 137 (2024), paper no. 108165. DOI: https://doi.org/10.1016/j.cnsns.2024.108165 DOI: https://doi.org/10.1016/j.cnsns.2024.108165

X. Tan and T. Chen, A mixed finite element approach for a variational-hemivariational inequality of incompressible Bingham fluids. J. Sci. Comput. 103 (2025), paper no. 36. DOI: https://doi.org/10.1007/s10915-025-02847-8 DOI: https://doi.org/10.1007/s10915-025-02847-8

R. Temam, Mathematical Problems in Plasticity. Gauthier-Villlars, Paris, 1985.

M. Ulbrich, Semismooth Newton Methods for Variational Inequalities and Constrained Optimization Problems in Function Spaces. SIAM, 2011. DOI: https://doi.org/10.1137/1.9781611970692

F. Wang and H. Dang, Randomized neural network methods for solving obstacle problems. Banach Center Publ. 127 (2024), 261--276. DOI: https://doi.org/10.4064/bc127-12

F. Wang and H. Qi, A discontinuous Galerkin method for an elliptic hemivariational inequality for semipermeable media. Applied Mathematics Letters 109 (2020), paper no. 106572. DOI: https://doi.org/10.1016/j.aml.2020.106572 DOI: https://doi.org/10.1016/j.aml.2020.106572

F. Wang, S. Shah and B. Wu, Discontinuous Galerkin methods for hemivariational inequalities in contact mechanics. J. Sci. Comp. 95 (2023), paper no. 87. DOI: https://doi.org/10.1007/s10915-023-02212-7 DOI: https://doi.org/10.1007/s10915-023-02212-7

F. Wang, B. Wu and W. Han, The virtual element method for general elliptic hemivariational inequalities. Journal of Computational and Applied Mathematics 389 (2021), paper no. 113330. DOI: https://doi.org/10.1016/j.cam.2020.113330 DOI: https://doi.org/10.1016/j.cam.2020.113330

F. Wang and J. Zhao, Conforming and nonconforming virtual element methods for a Kirchhoff plate contact problem. IMA J. Numer. Anal. 41 (2021), 1496-–1521. DOI: https://doi.org/10.1093/imanum/draa005 DOI: https://doi.org/10.1093/imanum/draa005

S. Wang, W. Xu, W. Han and W. Chen, Numerical analysis of history-dependent variational-hemivariational inequalities. Sci. China Math. 63 (2020), 2207–-2232. DOI: https://doi.org/10.1007/s11425-019-1672-4 DOI: https://doi.org/10.1007/s11425-019-1672-4

W. Wang, X.-L. Cheng and W. Han, Analysis and finite element solution of a Navier–Stokes hemivariational inequality for incompressible fluid flows with damping. Nonlinear Analysis: Real World Applications 87 (2026), paper no. 104439. DOI: https://doi.org/10.1016/j.nonrwa.2025.104439 DOI: https://doi.org/10.1016/j.nonrwa.2025.104439

P. Wriggers, Computational Contact Mechanics. Second edition, Springer, Berlin, 2006. DOI: https://doi.org/10.1007/978-3-540-32609-0

P. Wriggers, W. T. Rust and B. D. Reddy, A virtual element method for contact. Comput. Mech. 58 (2016), 1039--1050. DOI: https://doi.org/10.1007/s00466-016-1331-x DOI: https://doi.org/10.1007/s00466-016-1331-x

B. Wu, F. Wang and W. Han, Virtual element method for a frictional contact problem with normal compliance. Commun. Nonlinear Sci. Numer. Simul. 107 (2022), paper no. 106125. DOI: https://doi.org/10.1016/j.cnsns.2021.106125 DOI: https://doi.org/10.1016/j.cnsns.2021.106125

B. Wu, F. Wang and W. Han, The virtual element method for a contact problem with wear and unilateral constraint. Appl. Numer. Math. 206 (2024), 29--47. DOI: https://doi.org/10.1016/j.apnum.2024.08.004 DOI: https://doi.org/10.1016/j.apnum.2024.08.004

W. Xiao and M. Ling, The virtual element method for general variational-hemivariational inequalities with applications to contact mechanics. J. Comput. Appl. Math. 428 (2023), paper no. 115152. DOI: https://doi.org/10.1016/j.cam.2023.115152 DOI: https://doi.org/10.1016/j.cam.2023.115152

W. Xiao and M. Ling, A priori error estimate of virtual element method for a quasivariational-hemivariational inequality. Commun. Nonlinear Sci. Numer. Simul. 121 (2023), paper no. 107222. DOI: https://doi.org/10.1016/j.cnsns.2023.107222 DOI: https://doi.org/10.1016/j.cnsns.2023.107222

W. Xiao and M. Ling, Virtual element method for a history-dependent variational-hemivariational inequality in contact problems. J. Sci. Comput. 96 (2023), paper no. 82. DOI: https://doi.org/10.1007/s10915-023-02310-6 DOI: https://doi.org/10.1007/s10915-023-02310-6

W. Xu, Z. Huang, W. Han, W. Chen and C. Wang, Numerical approximation of an electro-elastic frictional contact problem modeled by hemivariational inequality. Comput. Appl. Math. 39 (2020), paper no. 265. DOI: https://doi.org/10.1007/s40314-020-01305-5 DOI: https://doi.org/10.1007/s40314-020-01305-5

W. Xu, Z. Huang, W. Han, W. Chen and C. Wang, Numerical analysis of history-dependent variational-hemivariational inequalities with applications in contact mechanics. J. Comput. Appl. Math. 351 (2019), 364--377. DOI: https://doi.org/10.1016/j.cam.2018.08.046 DOI: https://doi.org/10.1016/j.cam.2018.08.046

H. Xuan, X.-L. Cheng and L. Yuan, Numerical studies of a class of thermoviscoelastic frictional contact problem described by fractional differential hemivariational inequalities. J. Sci. Comput. 103 (2025), paper no. 4. DOI: https://doi.org/10.1007/s10915-025-02815-2 DOI: https://doi.org/10.1007/s10915-025-02815-2

I. Yousept, Maxwell quasi-variational inequalities in superconductivity. ESAIM Math. Model. Numer. Anal. 55 (2021), 1545--1568. DOI: https://doi.org/10.1051/m2an/2021028 DOI: https://doi.org/10.1051/m2an/2021028

E. Zeidler, Nonlinear Functional Analysis and its Applications. III: Variational Methods and Optimization. Springer-Verlag, New York, 1985.

Variational-Hemivariational Inequalities: A Brief Survey on Mathematical Theory and Numerical Analysis

Authors

DOI:

Keywords:

Abstract

References

Downloads

Published

Issue

Section

Categories

License

Make a Submission

Information

Browse

Browse Articles