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# A family of conforming mixed finite elements for linear elasticity on triangular grids

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### Abstract

This paper presents a family of mixed finite elements on triangular grids for solving the classical Hellinger-Reissner mixed problem of the elasticity equations. In these elements, the matrix-valued stress field is approximated by the full $$C^0$$-$$P_k$$ space enriched by $$(k-1)$$ $$H(\d)$$ edge bubble functions on each internal edge, while the displacement field by the full discontinuous $$P_{k-1}$$ vector-valued space, for the polynomial degree $$k\ge 3$$. The main challenge is to find the correct stress finite element space matching the full $$C^{-1}$$-$$P_{k-1}$$ displacement space. The discrete stability analysis for the inf-sup condition does not rely on the usual Fortin operator, which is difficult to construct. It is done by characterizing the divergence of local stress space which covers the $$P_{k-1}$$ space of displacement orthogonal to the local rigid-motion. The well-posedness condition and the optimal a priori error estimate are proved for this family of finite elements. Numerical tests are presented to confirm the theoretical results.

### Most cited references22

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### Finite element interpolation of nonsmooth functions satisfying boundary conditions

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### On the existence, uniqueness and approximation of saddle-point problems arising from lagrangian multipliers

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### A family of higher order mixed finite element methods for plane elasticity

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### Author and article information

###### Journal
2014-06-28
2015-01-20
###### Article
1406.7457