Math 673, Fall 2016: Numerical Methods for Variational Inequalities

Abner J. Salgado (asalgad1@utk.edu)

The purpose of this class will be to introduce the main ideas and techniques behind the analysis of finite element approximations of elliptic variational inequalities. The problems and techniques covered will range from classical ones (developed in the 1970’s) to recent research topics and results. A tentative list is:

- ∙
- Convex functions: convex sets and their separation, convex functions and their properties, polar functions, subdifferentials.
- ∙
- Minimization of convex functions and variational inequalities: existence, uniqueness of minimizers, characterization of solutions, examples.
- ∙
- Elements of duality: the primal problem and its dual, Lagrangians and saddle points, examples.

- ∙
- Variational inequalities in Hilbert space: variational inequalities in ℝ
^{n}, Lions – Stampacchia theorem. - ∙
- The classical and thin obstacle problems: existence, uniqueness and comments about regularity.

- ∙
- Review of finite elements: interpolation error, inverse inequalities, energy and L
^{2}error estimates for linear problems. - ∙
- Energy error estimates for variational inequalities.
- ∙
- L
^{∞}error estimates for linear problems (if necessary). - ∙
- L
^{∞}error estimates for variational inequalities: discrete maximum principle, M-matrices and the error estimate. - ∙
- Approximation of free boundaries: discrete and continuous nondegeneracy conditions, approximation in distance and measure of the free boundaries.

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- Descent methods.
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- Splitting methods.
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- Primal dual methods.
- ∙
- Semismooth Newton methods.

There will be no required textbook, some references are:

- 1.
- I. Ekeland, R. Teman. Convex analysis and variational problems.
- 2.
- D. Kinderlehrer, G. Stampacchia. An introduction to variational inequalities and their applications.
- 3.
- R. Glowinksi. Numerical methods for nonlinear variational problems.
- 4.
- R. Glowinksi, J.-L. Lions, R. Tremolieres. Numerical analysis of variational inequalities.
- 5.
- H. Brezis. Operateurs maximaux monotones et semi-groupes de contraction dans les espaces de Hilbert.
- 6.
- R.H. Nochetto, E. Otarola and A.J. Salgado. Convergence rates for the classical, thin and fractional elliptic obstacle problems. Philos. Trans. A 373 (2015), no. 2050, 20140449

Will be based on student presentations of selected topics. Possible topics:

- ∙
- The Signorini problem.
- ∙
- Multilevel methods for elliptic variational inequalities.
- ∙
- Higher order obstacle problems.
- ∙
- A posteriori error estimation for obstacle problems.
- ∙
- Quasivariational inequalities.
- ∙
- Gradient constraints.
- ∙
- The norm of the projection in a Hilbert space.