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Discretization of semicoercive variational inequalities

Summary

Consider the variational inequality:

$$Find \hat x \varepsilon K such that \beta (\hat x, x - \hat x) \geqslant \lambda (x - \hat x) for all x \varepsilon K$$

and its discretization:

$$Find x_h \varepsilon K_h such that \beta (x_h , x - x_h ) \geqslant \lambda (x - x_h ) for all x \varepsilon K_h .$$

Here, in a real reflexive separable Banach spaceX, β is a continuous bilinear form onX × X that is nonnegative on the diagonal,λ ∈ 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 >o of subspaces ofX describes a discretization scheme. Then under Glowinski's realistic assumptions on the approximation ofK by{K h } h > o—not requiring that\(K_h \subseteq K - \) we prove norm convergence,\(\lim _{h \to 0} \left\| {x_h - \hat x} \right\| = 0\), provided the solution\(\hat x\) is unique andβ satisfies a Gårding inequality: There exist a compact operatorT 1 :X→X * and a positive constant α such thatβ(x, x)+〈T 1 x, x〉 ⩾ α∥x∥ 2 for allx ∈ X.

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Gwinner, J. Discretization of semicoercive variational inequalities. Aeq. Math. 42, 72–79 (1991). https://doi.org/10.1007/BF01818479

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AMS (1980) subject classification

  • Primary 65N30
  • Secondary 49A29