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  • Articles  (129)
  • 65N30  (129)
  • Mathematics  (129)
  • 1
    Electronic Resource
    Electronic Resource
    Calculs de complexité relatifs à une méthode de dissection emboîtée (1985)
    Springer
    Numerische Mathematik 47 (1985), S. 175-190 
    Details
    ISSN: 0945-3245
    Keywords: AMS (MOS): 65F05 ; 65N30 ; CR: G1.3
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Description / Table of Contents: Summary A ≪nested dissection≫ ordering is given for solving any system of linear equationsAX=B, for the family ofsparse symmetric positive definite matrices corresponding to the class of undirected graphs withbounded degree, and satisfyinga $$\sqrt n $$ -separator theorem. If we compare the methods and results presented by Rose [9], our algorithm is more simple, but the complexity results are less general because of the restriction of bounded degree. Besides, our proofs use continually the arborescent structure on the family of separators, which is, by another way, a partition of the set of vertices for the initial graph. Then, the general implementation scheme in the finite element package MODULEF, fortwo-dimensional finite element problems, is presented, and numerical comparisons between our ordering and the standard envelope method are given.
    Notes: Résumé Nous présentons une numérotation de type ≪Nested Dissection≫ des inconnues d'un système linéaireAX=B pour des ensembles de matricescreuses symétriques définies positives correspondant à des famille de graphes non orientés,à degré borné, et admettant un $$\sqrt n $$ -thérème de séparation. Comparativement aux méthodes et résultats de Rose [9], l'algorithme présenté est plus simple, mais les théorèmes de complexité moins généraux, en raison de l'hypothèse restrictive de degré borné. En outre, les démonstrations font appel en permanence à la structure d'arbre sur la famille des séparateurs qui constitue, par ailleurs, une partition de l'ensemble des sommets du graphe initial. Nous présentons ensuite le schéma général d'implémentation dans le cadre du code d'éléments finis MODULEF pourdes problèmes plans d'éléments finis, et nous donnons quelques mesures comparatives avec la numérotation plus classique qui tend à minimiser le profil de la matrice.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF01389708
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  • 2
    Electronic Resource
    Electronic Resource
    Additive Schwarz methods for thep-version finite element method (1993)
    Springer
    Numerische Mathematik 66 (1993), S. 493-515 
    Details
    ISSN: 0945-3245
    Keywords: 65N55 ; 65N30
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Summary In this paper, we study some additive Schwarz methods (ASM) for thep-version finite element method. We consider linear, scalar, self adjoint, second order elliptic problems and quadrilateral elements in the finite element discretization. We prove a constant bound independent of the degreep and the number of subdomainsN, for the condition number of the ASM iteration operator. This optimal result is obtained first in dimension two. It is then generalized to dimensionn and to a variant of the method on the interface. Numerical experiments confirming these results are reported. As is the case for other additive Schwarz methods, our algorithms are highly parallel and scalable.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF01385709
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  • 3
    Electronic Resource
    Electronic Resource
    Fourier-Chebyshev pseudospectral method for two-dimensional vorticity equation (1993)
    Springer
    Numerische Mathematik 66 (1993), S. 329-346 
    Details
    ISSN: 0945-3245
    Keywords: 65N30 ; 76D99
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Summary A Fourier-Chebyshev pseudospectral scheme is proposed for two-dimensional unsteady vorticity equation. The generalized stability and convergence are proved strictly. The numerical results are presented.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF01385701
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  • 4
    Electronic Resource
    Electronic Resource
    Robustness of multi-grid applied to anisotropic equations on convex domains and on domains with re-entrant corners (1993)
    Springer
    Numerische Mathematik 66 (1993), S. 373-398 
    Details
    ISSN: 0945-3245
    Keywords: 65F10 ; 65N30
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Summary We analyse multi-grid applied to anisotropic equations within the framework of smoothing and approximation-properties developed by Hack busch. For a model anisotropic equation on a square, we give an up-till-now missing proof of an estimate concerning the approximation property which is essential to show robustness. Furthermore, we show a corresponding estimate for a model anisotropic equation on an L-shaped domain. The existing estimates for the smoothing property are not suitable to prove robustness for either 2-cyclic Gauss-Seidel smoothers or for less regular problems such as our second model equation. For both cases, we give sharper estimates. By combination of our results concerning smoothing- and approximation-properties, robustness of W-cycle multi-grid applied to both our model equations will follow for a number of smoothers.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF01385703
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  • 5
    Electronic Resource
    Electronic Resource
    Estimation of the effect of numerical integration in finite element eigenvalue approximation (1989)
    Springer
    Numerische Mathematik 56 (1989), S. 735-762 
    Details
    ISSN: 0945-3245
    Keywords: AMS(MOS): 65D30 ; 65N15 ; 65N25 ; 65N30 ; CR: G 1.8
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Summary Finite element approximations of the eigenpairs of differential operators are computed as eigenpairs of matrices whose elements involve integrals which must be evaluated by numerical integration. The effect of this numerical integration on the eigenvalue and eigenfunction error is estimated. Specifically, for 2nd order selfadjoint eigenvalue problems we show that finite element approximations with quadrature satisfy the well-known estimates for approximations without quadrature, provided the quadrature rules have appropriate degrees of precision.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF01405286
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  • 6
    Electronic Resource
    Electronic Resource
    On the finite volume element method (1990)
    Springer
    Numerische Mathematik 58 (1990), S. 713-735 
    Details
    ISSN: 0945-3245
    Keywords: AMS(MOS): 65N10 ; 65N30 ; CR: G1.8
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Summary The finite volume element method (FVE) is a discretization technique for partial differential equations. It uses a volume integral formulation of the problem with a finite partitioning set of volumes to discretize the equations, then restricts the admissible functions to a finite element space to discretize the solution. this paper develops discretization error estimates for general selfadjoint elliptic boundary value problems with FVE based on triangulations with linear finite element spaces and a general type of control volume. We establishO(h) estimates of the error in a discreteH 1 semi-norm. Under an additional assumption of local uniformity of the triangulation the estimate is improved toO(h 2). Results on the effects of numerical integration are also included.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF01385651
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  • 7
    Electronic Resource
    Electronic Resource
    The finite element method for nonlinear elliptic equations with discontinuous coefficients (1990)
    Springer
    Numerische Mathematik 58 (1990), S. 51-77 
    Details
    ISSN: 0945-3245
    Keywords: AMS(MOS) ; 65N30 ; CR ; G1.8
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Summary The study of the finite element approximation to nonlinear second order elliptic boundary value problems with discontinuous coefficients is presented in the case of mixed Dirichlet-Neumann boundary conditions. The change in domain and numerical integration are taken into account. With the assumptions which guarantee that the corresponding operator is strongly monotone and Lipschitz-continuous the following convergence results are proved: 1. the rate of convergenceO(h ε) if the exact solutionu∈H 1 (Ω) is piecewise of classH 1+ε (0〈ε≦1);2. the convergence without any rate of convergence ifu∈H 1 (Ω) only.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF01385610
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  • 8
    Electronic Resource
    Electronic Resource
    Numerical analysis of oscillations in nonconvex problems (1991)
    Springer
    Numerische Mathematik 59 (1991), S. 747-767 
    Details
    ISSN: 0945-3245
    Keywords: 65N30 ; CR: G1.8
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Summary We study numerically the pattern of the minimizing sequences of nonconvex problems which do not admit a minimizer.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF01385808
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  • 9
    Electronic Resource
    Electronic Resource
    Analyse d'un élément mixte pour le problème de Stokes (1992)
    Springer
    Numerische Mathematik 62 (1992), S. 149-160 
    Details
    ISSN: 0945-3245
    Keywords: 65N30
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Description / Table of Contents: Résumé Nous considérons quelques extensions des résultats classiques d'estimation d'erreur pour les problèmes mixtes. Nous étudions en particulier l'effet de l'emploi d'une approximation non conforme et celui de la décomposition de la formeb(·,·) enb 1(·,·) etb 2(·,·). Nous employons cette dćomposition pour affiner certaines estimations dans le cas où la formea(·,·) n'est pas coercive dans la bonne norme. Nous considérons aussi des résultats de dualité dans le cas non conforme.
    Notes: Summary We consider some extensions of the classical error estimates for mixed problems. We take into account, in particular, nonconforming approximations and the splitting of the bilinear formb(·,·), intob 1(·,·) andb 2(·,·). We use this last fact to sharpen some estimates in a case where the forma(·,·) does not have coercivity in the right norm. We also consider duality results in the nonconforming case.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF01396224
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  • 10
    Electronic Resource
    Electronic Resource
    Preconditioning nonconforming finite element methods for treating Dirichlet boundary conditions. I (1992)
    Springer
    Numerische Mathematik 62 (1992), S. 391-411 
    Details
    ISSN: 0945-3245
    Keywords: 65N22 ; 65N30
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Summary This work deals with theL 2 condition numbers and the distribution of theL 2 singular values of the preconditioned operators {B h −1 Ah}0〈h〈1, whereA h andB h are finite element discretizations of second order elliptic operators,A andB respectively. For conforming finite elements, it was shown in the work of Goldstein, Manteuffel and Parter that if the leading part ofB is a scalar multiple (1/Θ) of the leading part ofA, then the singular values ofB h −1 A h “cluster” and “fill-in” the interval [θ min,θ max], where 0〈θ min≦θ max are the minimum and maximum of the factor Θ. As a generalization of these results, the current work includes nonconforming finite element methods which deal with Dirichlet boundary conditions. It will be shown that, in this more general setting, theL 2 condition numbers of {B h −1 A h } are uniformly bounded. Moreover, the singular values also “cluster” and “fill-in” the same interval. In particular, if the leading part ofB is the same as the leading part ofA, then the singular values cluster about the point {1}. Two specific methods are given as applications of this theory. They are the penalty method of Babuška and the method of “nearly zero” boundary conditions of Nitsche.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF01396236
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