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Multigrid algorithms for variational inequalities. (English) Zbl 0628.65046

A variational inequality of the form \[ (1)\;(A_ h(u_ h),u_ h-v)\leq (f_ h,u_ h-v)\quad (\forall)\quad v\in K_ h=K^ 1_ h=\{v\in R^ N;\quad v\leq \psi^ 2_ h\} \] is written as a Hamilton-Jacobi-Bellman equation: \(\max [A_ h(u_ h)-f_ h,u_ h-\psi^ 2_ h]=0\) and an iterative procedure suggested by P. L. Lions and B. Mercier [RAIRO Anal. Numér. 14, 369-393 (1980; Zbl 0469.65041)] is proved to be monotonically convergent and to lead to a reduced algebraic system of the form: (2) \(\hat A^ n_ h(v_ h^{n+1})-\hat f^ n_ h=0\) at each iteration \(n=0,1,2,...\). Similar results are obtained for lower obstacle problems in which \(K_ h=K^ 2_ h=\{v\in R^ N;\quad v\geq \psi^ 1_ h\},\) for two-sided obstacle problems in which \(K_ h=K^ 3_ h=\{v\in R^ N;\quad \psi^ 1_ h\leq v\leq \psi^ 2_ h\}\) and for quasi-variational inequalities.
If the variational inequality (1) is the discretization of an elliptic variational inequality then the numerical solution of the reduced system (2) is obtained by a multigrid method with respect to a decreasing sequence of stepsizes, \(h_{k+1}=h_ k/2\), \(k=0,1,..\). m-1, \(h_ 0>0\) being given. Several multigrid procedure are proposed and numerical experiments for the elastic-plastic torsion problem of a cylindrical bar, the minimal surface over an obstacle and a stochastic inventory problem are presented.
Reviewer: S.Mirica

MSC:

65K10 Numerical optimization and variational techniques
65H10 Numerical computation of solutions to systems of equations
35J85 Unilateral problems; variational inequalities (elliptic type) (MSC2000)
49J40 Variational inequalities
90C33 Complementarity and equilibrium problems and variational inequalities (finite dimensions) (aspects of mathematical programming)

Citations:

Zbl 0469.65041
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