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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)

Citations:

Zbl 0536.65054
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References:

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