R. S. Falk, Error estimates for the approximation of a class of variational inequalities, Math. Comp. 28 (1974), 963-971.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....since the matrix B is not diagonal, and the latter method yields systems of linear equations of poor condition number. The finite element error estimates of order O(h 3=4 ) are obtained in [15] 14] and [18] for polyhedral or convex smooth domains Omega Gamma We also refer to the earlier works [10], 3] and [13] on the finite element methods and error analysis for general variational inequalities. The augmented Lagrangian methods have been widely used in nonlinear constrained optimization problems and nonlinear boundary value problems to relax complicated constraints or difficult couplings ....

R.S. Falk, Error estimates for the approximation of a class of variational inequalities, Math. Comp. 28 (1974), 963-971.

.... theorem, there exists a unique solution of (10) in H 1 0( 1 2 Variational inequalities arising from problems with unilateral constraints such as (5) and (10) are classically approximated by the nite element method, 25] 18] and [19] error estimates have been established in Falk [13], Mosco Strang [31] Glowinski Lions Tr emoli eres [18] Ciarlet [5] Glowinski [19] Brezzi Hager Raviart [3] 20] 4] and Falk Mercier [14] For the particular Signorini problem (i.e. Signorini boundary condition on the whole boundary) Brezzi HagerRaviart [3] use a piecewise linear nite ....

R. Falk, Error estimates for the approximation of a class of variational inequalities, Math. Comput, 28, 963-971 (1974).

....1] j v is linear on [x i ; x i 1 ]g. Setting K h = K V h the discrete version of (1.1) is u h 2 K h : a(u h ; u h ) f; u h ) 8 2 K h : 1. 2) A priori error analysis: The discussion starts with stating the result of an error estimate in the energy norm, which can be found, e.g. in Falk [7], ku u h k V ch : Using the de nitions B h = fx 2 f0; 1g j u h (x) 0g and G = fv 2 V jv 0 on B h and a(U u; v u h u) 0g, one can use a duality argument z 2 G : u u h ; z) a( z; z) 8 2 G ; 1.3) to obtain an improved L 2 estimate. U appearing in the de nition of G is the ....

R.S. Falk. Error estimates for the approximation of a class of variational inequalities. Math. Comp., 28:963-971, 1974.

....by (ru h ; r( u h ) f; u h ) 8 2 K h ; 3.3) where K h = fv 2 V h j v h in g. The nite dimensional problem can be shown to be uniquely solvable following the same line of arguments as in the continuous case. A priori error estimates in the energy norm can be found in, e.g. Falk [9] and Brezzi et al. 7] A posteriori error estimate In Natterer [14] there is described a generalisation of Nitsche s trick for variational inequalities. We employ a variant of his techniques to derive an a posteriori error estimate for the scheme (3.3) Motivated by (2.7) we would like to de ....

R.S. Falk. Error estimates for the approximation of a class of variational inequalities. Math. Comp., 28:963-971, 1974.

.... u) x)dx 8v 2 K h : 4) By Stampacchia s Theorem, Problem (4) has a unique solution. Indeed the set K h is non empty since the function min( P i = 1; N i i ; 0) belongs to K h . Error estimates for the approximate nite solution of the elliptic variational inequalities can be found in Falk [7], Mosco Strang [23] Glowinski Lions Tr emoli eres [12] Ciarlet [3] Brezzi Hager Raviart [2] and Falk Mercier [8] Error estimates of order 1 in the discretization step are known for the discretization of the obstacle problem using linear elements [7] 2] The monotony algorithm is derived on a ....

.... variational inequalities can be found in Falk [7] Mosco Strang [23] Glowinski Lions Tr emoli eres [12] Ciarlet [3] Brezzi Hager Raviart [2] and Falk Mercier [8] Error estimates of order 1 in the discretization step are known for the discretization of the obstacle problem using linear elements [7], 2] The monotony algorithm is derived on a strong formulation of Problem (4) which is easily shown to be equivalent to (4) Proposition 2.1 Let u be the unique solution to Problem (4) and let U = u 1 ; uN ) 2 IR N be de ned by u i = u(s i ) 8i 2 f1; Ng, then u is a solution ....

R. Falk, Error estimates for the approximation of a class of variational inequalities, Math. Comput, 28, 963-971, 1974.

.... theorem, there exists a unique solution of (10) in H 1 0( Omega Gamma1 2 Variational inequalities arising from problems with unilateral constraints such as (5) and (10) are classically approximated by the finite element method, 25] 18] and [19] error estimates have been established in Falk [13], Mosco Strang [31] Glowinski Lions Tr emoli eres [18] Ciarlet [5] Glowinski [19] Brezzi Hager Raviart [3] 20] 4] and Falk Mercier [14] For the particular Signorini problem (i.e. Signorini boundary condition on the whole boundary) Brezzi HagerRaviart [3] use a piecewise linear finite ....

R. Falk, Error estimates for the approximation of a class of variational inequalities, Math. Comput, 28, 963-971 (1974).

....provided that the bilinear form B( is symmetric and H(#) elliptic, i.e. # = 0 in (2.2) Corresponding to (2.4) 3.25) reduces to the finite dimensional approximate problem (3. 26) J (u s ) inf v #K s J (v s ) which, under suitable conditions on the approximate convex set K s [16], 18] can be solved by mathematical programming subject to a finite number of constraints induced by K s . Clearly, the variational error problem (2.7) suggests the needed formulation for error estimation; that is, determine e # K # c such that (3.27) B( e, w) G(w) # B( e, ....

R. S. FALK, Error estimates for the approximation of a class of variational inequalities, Math. Comp., 28 (1974), pp. 963--971.

....of convergence in terms of the mesh size h. The numerical convergence rates observed are in good agreement with the predicted ones. Mathematics Subject Classification (1991) 65K10, 65N30, 49J40 1. Introduction The aim of this paper is to extend the well known abstract error estimate of Falk [10] to the case of semicoercive variational inequalities. Problems of this kind occur e.g in contact problems of elasticity if one does not require a Dirichlet condition on the boundary. However, the situation is more involved than in the coercive case. The existence of a solution is only guaranteed ....

....of Numer. Math. 69: 103 116 (1994) W. Spann Remark. a) If NA span K = f0g and K h ae span K then the error estimate can be proven without the term inf v2K ku h Gamma vk 2 . In this case a h is uniformly positive definite for h sufficiently small and (3.12) 3. 13) can directly be proven [10, 4, 6, 15]. b) In the case K = V , K h = V h the error estimate is reduced to Strang s Lemma. c) Convergence can be proven without the assumption NA N(Au Gamma f) cone(K Gamma u) f0g ( 12] for a = a h , f = f h ) The proof of Theorem 3.1 will be given in several steps. Some easy consequences of ....

Falk, R.S. (1974): Error estimates for the approximation of a class of variational inequalities. Math. Comput. 28, 963--971

....and let K h = V h K; then K h is a non empty, closed, convex subset of V h . We may now pose the finite element approximation to problem (1. 22) Find u h 2 K h such that a(u h ; v h Gamma u h ) L(v h Gamma u h ) for all v h 2 K h : 1:28) We have the following result, due to Falk [20]. Theorem 1.6 Under the assumptions made on the data, and for regular refinements of piecewise linear finite elements, the approximate solution u h of the problem (1.28) to the problem (1.22) satisfies the error estimate jju Gamma u h jj V Ch; 1:29) 1 PRELIMINARIES 18 where C is a positive ....

R.S. Falk, Error estimates for the approximation of a class of variational inequalities, Math. Comp. 28 (1974) 963--971.

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R. S. Falk, Error estimates for the approximation of a class of variational inequalities, Math. Comp. 28 (1974), 963-971.

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