On Noncoercive Variational Inequalities
Abstract
We consider variational inequalities with different trial and test spaces and a possibly noncoercive bilinear form. Well-posedness is shown under general conditions that are, e.g., valid for the space-time variational formulation of parabolic variational inequalities. Moreover, we prove an estimate for the error of a Petrov--Galerkin approximation in terms of the residual. For parabolic variational inequalities the arising estimate is independent of the final time.
Keywords
MSC codes
Get full access to this article
View all available purchase options and get full access to this article.
References
1.
W. Arendt and K. Urban, Partielle Differenzialgleichungen, Spektrum Akademischer Verlag, Heidelberg, 2010.
2.
C. Bernardi, C. Canuto, and Y. Maday, Generalized inf-sup conditions for Chebyshev spectral approximation of the Stokes problem, SIAM J. Numer. Anal., 25 (1988), pp. 1237--1271.
3.
R. Dautray and J. L. Lions, Mathematical Analysis and Numerical Methods for Science and Technology, Volume 5, Evolution Problems I, Springer-Verlag, Berlin, 1992.
4.
L. C. Evans, Partial Differential Equations, Grad. Stud. Math. 19, AMS, Providence, RI, 2010.
5.
H.-G. Roos and C. Grossmann, Numerische Behandlung partieller Differentialgleichungen, 3rd ed., Teubner-Verlag, Stuttgart, 2005.
6.
B. Haasdonk, J. Salomon, and B. Wohlmuth, A Reduced Basis Method for Parametrized Variational Inequalities, SIAM J. Numer. Anal., 50 (2012), pp. 2656--2676.
7.
B. Haasdonk, J. Salomon, and B. Wohlmuth, A Reduced Basis Method for Parameterized Variational Inequalities, SimTech preprint 2011--17, University of Stuttgart, 2011.
8.
B. Haasdonk, J. Salomon, and B. Wohlmuth, A Reduced Basis Method for the Simulation of American Options, arXiv:1201.3289, 2012.
9.
K. Ito and K. Kunisch, Parabolic variational inequalities: The Lagrange multiplier approach, J. Math. Pures Appl. (9), 85 (2006), pp. 415--449.
10.
D. Kinderlehrer and G. Stampacchia, An Introduction to Variational Inequalities and Their Applications, SIAM, Philadelphia, 2000.
11.
J.-L. Lions and G. Stampacchia, Variational inequalities, Comm. Pure Appl. Math., 20 (1967), pp. 493--519.
12.
J. Nečas, Sur une méthode pour resoudre les équations aux dérivées partielles du type elliptique, voisine de la variationnelle, Ann. Sc. Norm. Super. Pisa, Sci. Fis. Mat. III. Ser. 16 (1962), pp. 305--326.
13.
R. H. Nochetto, K. G. Siebert, and A. Veeser, Theory of adaptive finite element methods: An introduction, in Multiscale, Nonlinear and Adaptive Approximation, R. DeVore and A. Kunoth, eds., Springer, Berlin, 2009, pp. 409--542.
14.
G. Rozza and A. T. Patera, Reduced Basis Approximation and A Posteriori Error Estimation for Parametrized Partial Differential Equations, Massachusetts Institute of Technology, Cambridge, MA, 2006; also available online from http://augustine.mit.eduhttp://augustine.mit.edu.
15.
M. Rudd and K. Schmitt, Variational inequalities of elliptic and parabolic type, Taiwanese J. Math., 6 (2002), pp. 287--322.
16.
C. Schwab and R. Stevenson, Space-time adaptive wavelet methods for parabolic evolution problems, Math. Comp., 78 (2009), pp. 1293--1318.
17.
K. Urban and A. T. Patera, A new error bound for reduced basis approximation of parabolic partial differential equations, C. R. Acad. Sci. Paris Ser. I, 350 (2012), pp. 203--207.
18.
K. Urban and A. T. Patera, An improved error bound for reduced basis approximation of linear parabolic problems, Math. Comput., 83 (2014), pp. 1599--1615.
Information & Authors
Information
Published In
Copyright
© 2014, Society for Industrial and Applied Mathematics.
History
Submitted: 17 June 2013
Accepted: 30 June 2014
Published online: 11 September 2014
Keywords
MSC codes
Authors
Metrics & Citations
Metrics
Citations
If you have the appropriate software installed, you can download article citation data to the citation manager of your choice. Simply select your manager software from the list below and click Download.
Cited By
- How regularization concepts interfere with (quasi-)brittle damage: a comparison based on a unified variational frameworkContinuum Mechanics and Thermodynamics, Vol. 34, No. 6 | 5 September 2022
- Deep solution operators for variational inequalities via proximal neural networksResearch in the Mathematical Sciences, Vol. 9, No. 3 | 4 June 2022
- A reduced basis method for parametrized variational inequalities applied to contact mechanicsInternational Journal for Numerical Methods in Engineering, Vol. 121, No. 6 | 17 December 2019
- Reduced Basis Isogeometric Mortar Approximations for Eigenvalue Problems in VibroacousticsModel Reduction of Parametrized Systems | 6 September 2017
- An empirical interpolation approach to reduced basis approximations for variational inequalitiesMathematical and Computer Modelling of Dynamical Systems, Vol. 22, No. 4 | 23 June 2016
- Reduced Basis Methods for Pricing Options with the Black--Scholes and Heston ModelsSIAM Journal on Financial Mathematics, Vol. 6, No. 1 | 6 August 2015
- Numerical Investigations of an Error Bound for Reduced Basis Approximations of Noncoercice Variational InequalitiesIFAC-PapersOnLine, Vol. 48, No. 1 | 1 Jan 2015
View Options
Get Access
- Access via your Institution
- Questions about how to access this content? Contact SIAM at [email protected].