Rectangular mixed elements for elasticity with weakly imposed symmetry condition

Awanou, Gerard

doi:10.1007/s10444-011-9240-1

Cited by 19 publications

(19 citation statements)

References 27 publications

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“…For simplicity, Dirichlet boundary conditions are specified on the entire boundary in all examples. In 3 dimensions we employ the BDM 1 × Q 0 × Q 0 triple of elements proposed by Awanou [11], which are the rectangular analogues of the lowest order Arnold-Falk-Winther simplicial elements [9]. In 2 dimensions we use BDM 1 × Q 0 × Q cts 1 , a modified triple of elements with continuous Q 1 space for rotation introduced in [3].…”

Section: Numerical Resultsmentioning

confidence: 99%

“…be any stable triple of spaces for linear elasticity with weakly imposed stress symmetry, such as the Amara-Thomas [1], PEERS [8], Stenberg [42], Arnold-Falk-Winther [7,9,11], or Cockburn-Gopalakrishnan-Guzman [15,26] families of elements. For all spaces div X h = V h and there exists a projection operator Π :…”

Section: Mfe Approximationmentioning

confidence: 99%

“…Since in all cases p ≤ k, the last two error terms are of the lowest order and balancing them results in the choice H = O(h p+1 m+1/2 ). For example, for the lowest order Arnold-Falk-Winther space on simplices [9] and its extensions to rectangles in two and three dimensions [11] or quadrilaterals [7], X h,i × V h,i × W h,i = BDM 1 × P 0 × P 0 , so k = 1 and l = p = 0. In this case, taking m = 2 and the asymptotic scaling H = O(h 2/5 ) provides optimal convergence rate O(h).…”

Section: 3mentioning

confidence: 99%

“…For spaces with optimal stress convergence, l ≤ p ≤ k, so balancing the last two terms results in the choice H = O(h l+1 m+1/2 ). For the lowest order spaces in [7,9,11] with k = 1 and l = p = 0, taking m = 2 and the asymptotic scaling H = O(h 2/5 ) provides superconvergence rate O(h 7/5 ). We further note that the above result is not useful for spaces with l = p − 1, in which case the bound (4.42) from Theorem 4.6, which does not depend on l, provides a better rate.…”

Section: Superconvergence For the Displacementmentioning

confidence: 99%

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Domain decomposition and multiscale mortar mixed finite element methods for linear elasticity with weak stress symmetry

Khattatov

Yotov

2019

ESAIM: M2AN

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Two non-overlapping domain decomposition methods are presented for the mixed finite element formulation of linear elasticity with weakly enforced stress symmetry. The methods utilize either displacement or normal stress Lagrange multiplier to impose interface continuity of normal stress or displacement, respectively. By eliminating the interior subdomain variables, the global problem is reduced to an interface problem, which is then solved by an iterative procedure. The condition number of the resulting algebraic interface problem is analyzed for both methods. A multiscale mortar mixed finite element method for the problem of interest on non-matching multiblock grids is also studied. It uses a coarse scale mortar finite element space on the non-matching interfaces to approximate the trace of the displacement and impose weakly the continuity of normal stress. A priori error analysis is performed. It is shown that, with appropriate choice of the mortar space, optimal convergence on the fine scale is obtained for the stress, displacement, and rotation, as well as some superconvergence for the displacement. Computational results are presented in confirmation of the theory of all proposed methods.

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Section: Numerical Resultsmentioning

confidence: 99%

Section: Mfe Approximationmentioning

confidence: 99%

Section: 3mentioning

confidence: 99%

Section: Superconvergence For the Displacementmentioning

confidence: 99%

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Domain decomposition and multiscale mortar mixed finite element methods for linear elasticity with weak stress symmetry

Khattatov

Yotov

2019

ESAIM: M2AN

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“…In this formulation the symmetry of the stress is imposed weakly using a Lagrange multiplier, which is a skew-symmetric matrix and has a physical meaning of rotation. Our first method, referred to as MSMFE-0, is based on the spaces BDM 1 × Q 0 × Q 0 developed in [6,9], using BDM 1 stress and piecewise constant displacement and rotation. The BDM 1 space has two normal degrees of freedom per edge, which can be associated with the two vertices.…”

Section: Introductionmentioning

confidence: 99%

A Multipoint Stress Mixed Finite Element Method for Elasticity on Simplicial Grids

Ambartsumyan¹,

Khattatov²,

Nordbotten³

et al. 2020

SIAM J. Numer. Anal.

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We develop a multipoint stress mixed finite element method for linear elasticity with weak stress symmetry on quadrilateral grids, which can be reduced to a symmetric and positive definite cell centered system. The method is developed on simplicial grids in [4]. The method utilizes the lowest order Brezzi-Douglas-Marini finite element spaces for the stress and the trapezoidal quadrature rule in order to localize the interaction of degrees of freedom, which allows for local stress elimination around each vertex. We develop two variants of the method. The first uses a piecewise constant rotation and results in a cell-centered system for displacement and rotation. The second uses a continuous piecewise bilinear rotation and trapezoidal quadrature rule for the asymmetry bilinear form. This allows for further elimination of the rotation, resulting in a cell-centered system for the displacement only. Stability and error analysis is performed for both methods. First-order convergence is established for all variables in their natural norms. A duality argument is employed to prove second order superconvergence of the displacement at the cell centers. Numerical results are presented in confirmation of the theory.

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A multipoint stress mixed finite element method for elasticity on quadrilateral grids

Ambartsumyan

Khattatov

Nordbotten

et al. 2020

Numerical Methods Partial

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We develop a multipoint stress mixed finite element method for linear elasticity with weak stress symmetry on quadrilateral grids, which can be reduced to a symmetric and positive definite cell centered system. The method utilizes the lowest order Brezzi-Douglas-Marini finite element spaces for the stress and the trapezoidal quadrature rule in order to localize the interaction of degrees of freedom, which allows for local stress elimination around each vertex. We develop two variants of the method. The first uses a piecewise constant rotation and results in a cell-centered system for displacement and rotation. The second uses a continuous piecewise bilinear rotation and trapezoidal quadrature rule for the asymmetry bilinear form. This allows for further elimination of the rotation, resulting in a cell-centered system for the displacement only. Stability and error analysis is performed for both methods. First-order convergence is established for all variables in their natural norms. A duality argument is employed to prove second order superconvergence of the displacement at the cell centers. Numerical results are presented in confirmation of the theory. KEYWORDS elasticity, mixed finite element, multipoint stress 1 INTRODUCTION Stress-displacement mixed finite element (MFE) elasticity formulations have been studied extensively due to their local momentum conservation with continuous normal stress and locking-free approximation, see [1] and references therein. These methods result in saddle point type algebraic

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Rectangular mixed elements for elasticity with weakly imposed symmetry condition

Cited by 19 publications

References 27 publications

Domain decomposition and multiscale mortar mixed finite element methods for linear elasticity with weak stress symmetry

Domain decomposition and multiscale mortar mixed finite element methods for linear elasticity with weak stress symmetry

A Multipoint Stress Mixed Finite Element Method for Elasticity on Simplicial Grids

A multipoint stress mixed finite element method for elasticity on quadrilateral grids

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