## 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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