Complementarity systems and approximation of variational inequalities

Mosco, Umberto; Scarpini, F.

doi:10.1051/m2an/197509r100831

Cited by 12 publications

(6 citation statements)

References 20 publications

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“…A different approach dealing directly with the contact set is due to U. Mosco and F. Scarpini [24], and is based on reduction of (VI) to a complementarity system.…”

Section: Numerical Solution Of the Obstacle Problem 47mentioning

confidence: 99%

Shape optimization approach to numerical solution of the Obstacle Problem

Bogomolny

Hou

1984

Appl Math Optim

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Abstract. We discuss an algorithm for the numerical solution of the Obstacle Problem in which the coincidence set is considered as the prime unknown. Domain functionals are defined for which the coincidence set serves as the minimizing element. Their gradients are computed (in the sense of the material derivative), and the gradient descent method employed to minimize these functionals. Numerical example is given.

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“…A different approach dealing directly with the contact set is due to U. Mosco and F. Scarpini [24], and is based on reduction of (VI) to a complementarity system.…”

Section: Numerical Solution Of the Obstacle Problem 47mentioning

confidence: 99%

Shape optimization approach to numerical solution of the Obstacle Problem

Bogomolny

Hou

1984

Appl Math Optim

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“…The minimization of I over K h -in other words, the computation of u h -is numerically not a difficult problem (cf. [5]). …”

Section: I(v) = A(v 9 V) -2(f V) = \\ (Vl + Vl -2fv) DX Dy Amentioning

confidence: 99%

One-sided approximation and variational inequalities

Mosco¹,

Strang²

1974

Bull. Amer. Math. Soc.

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ABSTRACT. For piecewise linear approximation of the unilateral Laplace equation (also known as the obstacle problem, and governed by a variational inequality), we prove that the gradient of the error u-u h is of order h. The proof rests on approximation of nonnegative functions U by nonnegative splines V h^U .We are interested in one of the first and most fundamental of the variational problems introduced by Fichera, Stampacchia, andFind that function u in the convex setIf the "obstacle function" xp were absent, this would be the classical Dirichlet problem for Poisson's equation -Au=f, and the condition for a minimum would be a variational equation: a(u,v) = (f,v) for v in 34? J. This is the weak form of Poisson's equation, and coincides with the engineer's "equation of virtual work". For minimization over K instead of the full space 3^J, the variational equation turns into an inequality-just as, for minimization of a function g over O^x^l, the possibility of minima at the endpoints alters the usual dg/dx=0. The condition that u be minimizing is (1) a (u, v -uSuppose we solve this problem approximately, by the Ritz principle: The approximation u h minimizes the functional I over a finite-dimensional AMS (MOS) subject classifications (1970). Primary 35J20, 65N30, 41A15.

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“…The existence and uniqueness of the solution of such a problem is now a well-known result, see [7]. The problem can be also formulated as a linear complementarity problem with a generalized Laplace operator, see [13]. We shall use the five-point discretization of the Laplace operator, which is equivalent to an approximation by special linear finite elements.…”

Section: Ilxlh = ( Fu (Ixl2 + L @~Xl2 + L O2xl2 ) Dtx Dt2 ) 1/2mentioning

confidence: 99%

A multilevel iterative method for symmetric, positive definite linear complementarity problems

Mandel

1984

Appl Math Optim

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Abstract. A fast iterative method for the solution of large, sparse, symmetric, positive definite linear complementarity problems is presented. The iterations reduce to linear iterations in a neighborhood of the solution if the problem is nondegenerate. The variational setting of the method guarantees global convergence.As an application, we consider a discretization of a Dirichlet obstacle problem by triangular linear finite elements. In contrast to usual iterative methods, the observed rate of convergence does not deteriorate with step size.

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Complementarity systems and approximation of variational inequalities

Cited by 12 publications

References 20 publications

Shape optimization approach to numerical solution of the Obstacle Problem

Shape optimization approach to numerical solution of the Obstacle Problem

One-sided approximation and variational inequalities

A multilevel iterative method for symmetric, positive definite linear complementarity problems

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