New mixed finite elements for plane elasticity and Stokes equations
 XiaoPing Xie,
 JinChao Xu
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We consider mixed finite elements for the plane elasticity system and the Stokes equation. For the unmodified HellingerReissner 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 stressvelocitypressure 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.
 Title
 New mixed finite elements for plane elasticity and Stokes equations
 Journal

Science China Mathematics
Volume 54, Issue 7 , pp 14991519
 Cover Date
 20110701
 DOI
 10.1007/s1142501141713
 Print ISSN
 16747283
 Online ISSN
 18691862
 Publisher
 SP Science China Press
 Additional Links
 Topics
 Keywords

 mixed finite element
 nonconforming
 elasticity
 Stokes equation
 65N15
 65N30
 65N50
 Industry Sectors
 Authors

 XiaoPing Xie ^{(1)}
 JinChao Xu ^{(2)}
 Author Affiliations

 1. School of Mathematics, Sichuan University, Chengdu, 610064, China
 2. Department of Mathematics, The Pennsylvania State University, University Park, PA, 16802, USA