On a Second-Order Multipoint Flux Mixed Finite Element Methods on Hybrid Meshes

Egger, Herbert; Radu, Bogdan

doi:10.1137/19m1236862

Cited by 3 publications

(3 citation statements)

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“…A third extension is to develop higher order methods, which requires constructing suitable general order family of stress spaces and quadrature rules. Such construction has been done for the MFMFE method on quadrilaterals and hexahedra in [31], see also [32] for second-order MFMFE methods on hybrid grids. A fourth possible extension is to develop these methods on polytopal grids using mimetic finite differences (MFD) or mixed virtual element methods (VEM).…”

Section: Discussionmentioning

confidence: 99%

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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Section: Discussionmentioning

confidence: 99%

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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“…In [3], the authors used the BDM 1 -P 0 pair together with the vertex rule to obtain first order elements. In [2], we managed to extend the theory and introduce corresponding second order elements. On simplices, the second order space coincides with the well-known pair RT 1 -P 1 .…”

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confidence: 99%

“…The two methods both yield first order convergence for the velocity and second order convergence for the projected pressure. For second order elements, we compare the lumped RT 1 -P 1 method from [2] with the hybridized triple RT d 1 -P 1 -P 1 . For these methods, we obtain second order convergence for the velocity and third order convergence for the projected pressure.…”

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confidence: 99%