By Blum H., Braess D., Suttmeier F.T.

While classical multigrid equipment are utilized to discretizations of variational inequalities, numerous issues are usually encountered generally as a result loss of easy possible limit operators. those problems vanish within the program of the cascadic model of the multigrid strategy which during this experience yields higher merits than within the linear case. in addition, a cg-method is proposed as smoother and as solver on coarse meshes. The potency of the recent set of rules is elucidated via try out calculations for a disadvantage challenge and for a Signorini challenge.

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**Example text**

5 Let K C E be a closed convex cone and let F be a •&logarithmically homogeneous barrier for K. Let E(K) be the maximal subspace of E contained in K. Then F is constant at each set of the form x+E(K), x € int K. If F is normal, then also D2F(x)[h, h] ± 0 for each h € E\E(K). Proof. Let h € E(K) and x e intK. We wish to prove that L>F[x][/i] = 0. Otherwise, we can assume that DF(x)[h] > 0 (replacing, if necessary, h by —h). Let is well defined and continuously differentiable on the half-axis {s > 0}.

Moreover, (Q*, F*)* = (Q, F) (as usually, the space (E*)* is identified with E). Proof. Let Q' = {£ € E* | the function F^(x) = F(x) — (£,x) is below bounded on Q}. We will prove that Q' = Q*. Since the inclusion Q* C Q' is evident, we have to establish the inverse inclusion. Let £ 6 Q', so that F^(x) is below bounded on Q. 1(ii)). 3(i)), which means that £ £ $>(Q)Since D2F(x) is nondegenerate for x € Q, Q* is open, while Q' is clearly a convex set. Thus, the set Q* is nonempty, open, and convex.

For this purpose, let us introduce some notation. For a closed convex domain G C E, let 7£(G) be the recessive cone of G Also, let be the cone anti-dual to K(G). Note that, if G does not contain any straight line, then 7£*(G) is a closed convex cone with a nonempty interior in E*. 2 Let G be a closed convex domain in E that does not contain any straight line and let F be a ^-self-concordant barrier for G. Then the Legendre transformation F* of F is defined precisely on the interior ofR*(G) and satisfies the following relations: (a) F* is strongly 1-self-concordant on miK*(G) and D2F*(s) is nondegenerate, s € int7£*(G); (b) sup{D2F*(s)[s,s] | s e intft*(G)} < tf; (c) The support function of G satisfies the inequality Proof.

### A cascadic multigrid algorithm for variational inequalities by Blum H., Braess D., Suttmeier F.T.

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