A nonconforming finite element method for a three-dimensional problem of the elasticity theory

Калинкин, А. М.; Laevsky, Yu. M.

doi:10.1163/156939806777973427

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“…It is shown in [5] that for mes 2´ΓD µ 0 bilinear form (1.15) generates inner products in spaces (1.13) and (1.14) and, hence, the functional…”

Section: Nonconforming Approximation Of the 3d Elasticity Problemmentioning

confidence: 99%

“…We study the method theoretically and experimentally.In [5], the authors proposed a nonconforming finite element method for a 3D problem of the elasticity theory in displacements, in which the degrees of freedom are related to cell faces on a parallelepiped grid. We construct a preconditioner based on the transition from an elasticity operator to the grid Laplace operator, the diagonalization of the matrix for tangential displacements, and the inner Chebyshev procedures for normal displacements.…”

mentioning

confidence: 99%

“…We construct a preconditioner based on the transition from an elasticity operator to the grid Laplace operator, the diagonalization of the matrix for tangential displacements, and the inner Chebyshev procedures for normal displacements. This becomes possible due to the Korn grid inequality from [5] and to the continuity of the grid inner energy product, which is established in Section 2. This paper is the continuation of this research involving the development of an iterative method for solving the corresponding grid system.…”

mentioning

confidence: 99%

“…In [5], the authors proposed a nonconforming finite element method for a 3D problem of the elasticity theory in displacements, in which the degrees of freedom are related to cell faces on a parallelepiped grid. This paper is the continuation of this research involving the development of an iterative method for solving the corresponding grid system.…”

mentioning

confidence: 99%

“…This becomes possible due to the Korn grid inequality from [5] and to the continuity of the grid inner energy product, which is established in Section 2. The first stage is the transition from the Lame operator to the Laplace operator.…”

mentioning

confidence: 99%

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Preconditioning of grid Lame equations in the nonconforming finite element method

Калинкин

Laevsky

2007

Russian Journal of Numerical Analysis and Mathematical Modelling

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In this paper a two-step iterative method for solving the corresponding grid system is proposed for nonconforming finite element approximations of a 3D problem in the elasticity theory. We construct a preconditioner based on the transition from an elasticity operator to the grid Laplace operator, the diagonalization of the matrix for tangential displacements, and the inner Chebyshev procedures for normal displacements. We study the method theoretically and experimentally.In [5], the authors proposed a nonconforming finite element method for a 3D problem of the elasticity theory in displacements, in which the degrees of freedom are related to cell faces on a parallelepiped grid. This paper is the continuation of this research involving the development of an iterative method for solving the corresponding grid system. We should first of all mention the papers [1, 2, 6] whose problems are close to those studied in the given paper and which consider two-level preconditioners for 2D problems of the elasticity theory. Unlike the above papers, the grid problem studied here has a stabilizing term, similarly to [4].The general scheme for constructing the preconditioner proposed in this paper consists of three stages. The first stage is the transition from the Lame operator to the Laplace operator. This becomes possible due to the Korn grid inequality from [5] and to the continuity of the grid inner energy product, which is established in Section 2. The second stage diagonalizes some blocks of the stiffness matrix corresponding to the Laplace operator. This makes it possible to substantially simplify further calculations. A similar approach is used in [1,2]. The equivalence of such modified matrix to the matrix corresponding to the Laplace operator is shown in Section 3. Finally, the third stage replaces the inversion of the Schur complement for the normal components of the displacement vector by some fixed number of steps of the Chebyshev process with an inner preconditioner. We follow here the paper [8] which, in particular, shows the relation between the condition number for the inner Chebyshev procedure and the common preconditioned system (Lemma 4.1). The preconditioner for the Schur complement is constructed in one of the forms of the SSOR method (in the form of the alternately triangular method in Samarsky's terminology) with an optimal parameter [10]. However, the above ap-£ Institute of Computational Mathematics and Mathematical Geophysics, Siberian Branch of the Russian Academy of Sciences, Novosibirsk 630090, RussiaThe work was supported by the Russian Foundation for Basic Research (04-01-00171) and the Russian-Dutch NWO-RFBR Program (047.008.007).Brought to you by | Florida International University Libraries Authenticated Download Date | 7/6/15 4:21 AM

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“…It is shown in [5] that for mes 2´ΓD µ 0 bilinear form (1.15) generates inner products in spaces (1.13) and (1.14) and, hence, the functional…”

Section: Nonconforming Approximation Of the 3d Elasticity Problemmentioning

confidence: 99%

mentioning

confidence: 99%

mentioning

confidence: 99%

mentioning

confidence: 99%

mentioning

confidence: 99%