TY - JOUR

T1 - New mixed finite elements for plane elasticity and Stokes equations

AU - Xie, Xiao Ping

AU - Xu, Jin Chao

N1 - Funding Information:
Acknowledgements This work was supported in part by National Natural Science Foundation of China (Grant No. 10771150), the National Basic Research Program of China (Grant No. 2005CB321701), and the program for New Century Excellent Talents in Universities (Grant No. NCET-07-0584). The first author would like to acknowledge the support during his visit to the Center for Computational Mathematics and Applications of Penn State. The authors also cordially thank the referees for their careful reading and helpful comments.

PY - 2011/7

Y1 - 2011/7

N2 - We consider mixed finite elements for the plane elasticity system and the Stokes equation. For the unmodified Hellinger-Reissner formulation of elasticity in which the stress and displacement fields are the primary unknowns, we derive two new nonconforming mixed finite elements of triangle type. Both elements use piecewise rigid motions to approximate the displacement and piecewise polynomial functions to approximate the stress, where no vertex degrees of freedom are involved. The two stress finite element spaces consist respectively of piecewise quadratic polynomials and piecewise cubic polynomials such that the divergence of each space restricted to a single simplex is contained in the corresponding displacement approximation space. We derive stability and optimal order approximation for the elements. We also give some numerical results to verify the theoretical results. For the Stokes equation, introducing the symmetric part of the gradient tensor of the velocity as a stress variable, we present a stress-velocity-pressure field Stokes system. We use some plane elasticity mixed finite elements, including the two elements we proposed, to approximate the stress and velocity fields, and use continuous piecewise polynomial functions to approximate the pressure with the gradient of the pressure approximation being in the corresponding velocity finite element spaces. We derive stability and convergence for these methods.

AB - We consider mixed finite elements for the plane elasticity system and the Stokes equation. For the unmodified Hellinger-Reissner formulation of elasticity in which the stress and displacement fields are the primary unknowns, we derive two new nonconforming mixed finite elements of triangle type. Both elements use piecewise rigid motions to approximate the displacement and piecewise polynomial functions to approximate the stress, where no vertex degrees of freedom are involved. The two stress finite element spaces consist respectively of piecewise quadratic polynomials and piecewise cubic polynomials such that the divergence of each space restricted to a single simplex is contained in the corresponding displacement approximation space. We derive stability and optimal order approximation for the elements. We also give some numerical results to verify the theoretical results. For the Stokes equation, introducing the symmetric part of the gradient tensor of the velocity as a stress variable, we present a stress-velocity-pressure field Stokes system. We use some plane elasticity mixed finite elements, including the two elements we proposed, to approximate the stress and velocity fields, and use continuous piecewise polynomial functions to approximate the pressure with the gradient of the pressure approximation being in the corresponding velocity finite element spaces. We derive stability and convergence for these methods.

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U2 - 10.1007/s11425-011-4171-3

DO - 10.1007/s11425-011-4171-3

M3 - Article

AN - SCOPUS:79960093697

VL - 54

SP - 1499

EP - 1519

JO - Science China Mathematics

JF - Science China Mathematics

SN - 1674-7283

IS - 7

ER -