Preconditioning discrete approximations of the Reissner-Mindlin plate model

Arnold, Douglas N.; Falk, Richard S.; Winther, Ragnar

doi:10.1051/m2an/1997310405171

Cited by 53 publications

(55 citation statements)

References 26 publications

(43 reference statements)

Supporting

Mentioning

Contrasting

Order By: Relevance

“…This method has the disadvantage that the stiffness matrix of mixed form, not symmetric and positively definite as the engineering community is used to. The same holds for the multigrid methods analyzed in the paper by Arnold, Falk and Winter [1] and Brenner [7].…”

Section: Introductionmentioning

confidence: 56%

Multigrid Methods for a Stabilized Reissner–Mindlin Plate Formulation

Schöberl¹,

Stenberg²

2009

SIAM J. Numer. Anal.

View full text Add to dashboard Cite

Abstract:We consider a stabilized finite element formulation for the Reissner-Mindlin plate bending model. The method, introduced in [18] uses standard bases functions for the deflection and rotation vector. Due to the stabilization the conditioning of the method is such that multigrid algorithms can readily been used. In the paper we first prove some error estimates needed for multigrid methods. Then we prove the a simple multigrid method has optimal complexity. Numerical results are also give.

show abstract

Section: Introductionmentioning

confidence: 56%

Multigrid Methods for a Stabilized Reissner–Mindlin Plate Formulation

Schöberl¹,

Stenberg²

2009

SIAM J. Numer. Anal.

View full text Add to dashboard Cite

show abstract

“…[17]). Let U h −1 and U h 0 be the respective finite dimensional subspaces of H 0 (div; Ω) 2 and W ×W , which will be specified in the subsequent section. Define…”

Section: Discrete Least-squares Functionals Let T H Be a Partition Omentioning

confidence: 99%

“…Let U h be a finite-dimensional subspace ofŨ. Assume that it satisfies the following approximation property: there exist a constant C and an integer γ 2 …”

Section: Remark 52 Similarly We Can Define the Discrete Counterparmentioning

confidence: 99%

See 1 more Smart Citation

Least Squares for the Perturbed Stokes Equations and the Reissner--Mindlin Plate

Cai¹

2000

SIAM J. Numer. Anal.

View full text Add to dashboard Cite

Abstract. In this paper, we develop two least-squares approaches for the solution of the Stokes equations perturbed by a Laplacian term. (Such perturbed Stokes equations arise from finite element approximations of the Reissner-Mindlin plate.) Both are two-stage algorithms that solve first for the curls of the rotation of the fibers and the solenoidal part of the shear strain, then for the rotation itself (if desired). One approach uses L 2 norms and the other approach uses H −1 norms to define the least-squares functionals. It is shown that the H −1 norm approach, under general assumptions, and the L 2 norm approach, under certain H 2 regularity assumptions, admit optimal performance for standard finite element discretization and either standard multigrid solution methods or preconditioners. These methods do not degrade when the perturbed parameter (the plate thickness) approaches zero. We also develop a three-stage least-squares method for the Reissner-Mindlin plate, which first solves for the curls of the rotation and the shear strain, next for the rotation itself, and then for the transverse displacement.

show abstract

“…In Section 4 we present some numerical experiments illustrating the computational efficiency This means that the L 2 norm of ω h is approximated so that of our inexact constraint preconditioning approach. As an alternative approach, optimal preconditioning methods based on multigrid ideas have recently been derived by Arnold, Falk and Winther [1] in the context of stable finite element discretisation methods of the Reissner-Mindlin plate model. When taking the limit of an infinitely thin plate -so that the Reissner-Mindlin plate model reduces to the biharmonic problem -the results in Section 4 are just as impressive as the numerical results in [1].…”

Section: Introductionmentioning

confidence: 99%

A Black-Box Multigrid Preconditioner for the Biharmonic Equation

Silvester

Mihajlović

2004

BIT Numerical Mathematics

View full text Add to dashboard Cite

Abstract.We examine the convergence characteristics of a preconditioned Krylov subspace solver applied to the linear systems arising from low-order mixed finite element approximation of the biharmonic problem. The key feature of our approach is that the preconditioning can be realized using any "black-box" multigrid solver designed for the discrete Dirichlet Laplacian operator. This leads to preconditioned systems having an eigenvalue distribution consisting of a tightly clustered set together with a small number of outliers. Numerical results show that the performance of the methodology is competitive with that of specialized fast iteration methods that have been developed in the context of biharmonic problems. (2000): 65F10, 65N12, 65N22, 65N55. AMS subject classification

show abstract

Preconditioning discrete approximations of the Reissner-Mindlin plate model

Cited by 53 publications

References 26 publications

Multigrid Methods for a Stabilized Reissner–Mindlin Plate Formulation

Multigrid Methods for a Stabilized Reissner–Mindlin Plate Formulation

Least Squares for the Perturbed Stokes Equations and the Reissner--Mindlin Plate

A Black-Box Multigrid Preconditioner for the Biharmonic Equation

Contact Info

Product

Resources

About