## Discretization of semicoercive variational inequalities.(English)Zbl 0739.65058

The convergence result of R. Glowinski [Numerical Methods for nonlinear variational problems. (1984; Zbl 0536.65054), Theorem I.5.2] on the discretization of variational inequalities is extended to the semicoercive case.
Author’s summary: Consider the variational inequality: Find $$\hat x \in K$$ such that $$\beta(\hat x,x-\hat x)\geq\lambda(x-\hat x)$$ for all $$x\in K$$ and its discretization: Find $$x_ h \in K_ h$$ such that $$\beta(x_ h,x-x_ h)\geq \lambda(x-x_ h)$$ for all $$x\in K_ h$$.
Here, in a real reflexive separable Banach space $$X$$, $$\beta$$ is a continuous bilinear form on $$X\times X$$ that is nonnegative on the diagonal, $$\lambda \in X^*$$ is a continuous linear form, and $$K\subseteq X$$, $$K_ h\subseteq X_ h$$ are closed convex nonvoid sets, where the family $$\{X_ h\}_{h>0}$$ of subspaces of $$X$$ describes a discretization scheme. Then under Glowinski’s realistic assumptions on the approximation of $$K$$ by $$\{ K_ h\}_{h>0}$$ — not requiring that $$K_ h\subseteq K$$ — we prove norm convergence, $$\lim_{h\to 0}\| x_ h-\hat x\| = 0$$, provided the solution $$\hat x$$ is unique and $$\beta$$ satisfies a Gårding inequality:
There exist a compact operator $$T_ 1: X\to X^*$$ and a positive constant $$\alpha$$ such that $$\beta(x,x)+\langle T_ 1x,x\rangle \geq \alpha\| x\|^ 2$$ for all $$x\in X$$.

### MSC:

 65K10 Numerical optimization and variational techniques 65N30 Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs 49J40 Variational inequalities 35J85 Unilateral problems; variational inequalities (elliptic type) (MSC2000)

Zbl 0536.65054
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### References:

 [1] Baiocchi, C., Gastaldi, F. andTomarelli, F.,Some existence results on noncoercive variational inequalities. Ann. Scuola Norm. Sup. Pisa CI. Sci. (4)13 (1986), 617–659. · Zbl 0644.49004 [2] Ciarlet, P. G.,The finite element method for elliptic problems. North-Holland, Amsterdam, 1978. · Zbl 0383.65058 [3] Costabel, M.,Starke Elliptizität von Randintegraloperatoren erster Art. Habilitationsschrift. THD-Reprint 868, Technische Hochschule Darmstadt, 1984. [4] Costabel, M. andWendland, W. L.,Strong ellipticity of boundary integral operators. J. Reine Angew. Math.372 (1986), 34–63. · Zbl 0628.35027 [5] Fenchel, W.,Über konvexe Funktionen mit vorgeschriebenen Niveaumannigfaltigkeiten. Math. Z.63 (1956), 496–506. · Zbl 0074.28902 [6] Fichera, G.,Boundary value problems of elasticity with unilateral constraints. In:Handbuch der Physik–Encyclopedia of Physics (S. Flügge; ed.) Band VI a/2 Festköpermechanik II. Springer, Berlin, 1972, pp. 391–424. [7] Friedman, A.,Partial differential equations. Holt, Rinehart and Winston, New York, 1969. · Zbl 0224.35002 [8] Glowinski, R.,Numerical methods for nonlinear variational problems. Springer, New York, 1984. · Zbl 0536.65054 [9] Gwinner, J.,Convergence and error analysis for variational inequalities and unilateral boundary value problems, Habilitationsschrift. THD-Preprint 1257, Technische Hochschule Darmstadt, 1989. · Zbl 0703.49012 [10] Gwinner, J.,Finite-element convergence for contact problems in plane linear elastostatics, to appear in Q. Appl. Math. · Zbl 0743.73025 [11] Hildebrandt, S. andWienholtz, E.,Constructive proofs of representativn theorems in separable Hilbert space. Comm. Pure Appl. Math.17 (1964), 369–373. · Zbl 0131.13401 [12] Kinderlehrer, D. andStampacchia, G.,An introduction to variational inequalities and their applications. Academic Press, New York, 1980. · Zbl 0457.35001 [13] Kohn, J. J. andNirenberg, L.,An algebra of pseudo-differential operators. Comm. Pure Appl. Math.18 (1965), 269–305. · Zbl 0171.35101 [14] Michlin, S. G.,Variationsmethoden der Mathematischen Physik. Akademie-Verlag, Berlin, 1962. · Zbl 0098.36909 [15] Minty, G. J.,Monotone (non linear) operators in Hilbert space. Duke Math. J.29 (1962), 341–346. · Zbl 0111.31202 [16] Stampacchia, G.,Variational inequalities. In:Theory and applications of monotone operators (A. Ghizetti; ed.). Edizione Oderisi, Gubbio, 1969, pp. 101–192.
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