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Stabilized mixed finite element methods for linear elasticity on simplicial grids in R^n

Long Chen, Jun Hu, Xuehai Huang

Comput. Methods Appl. Math. 2016

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ABSTRACT:

In this paper, we design two classes of stabilized mixed finite
element methods for linear elasticity on simplicial grids.  In the
first class of elements, we use $\boldsymbol{H}(\mathbf{div}, \Omega;
\mathbb{S})$-$P_k$ and $\boldsymbol{L}^2(\Omega;
\mathbb{R}^n)$-$P_{k-1}$ to approximate the stress and displacement
spaces, respectively, for $1\leq k\leq n$, and employ a stabilization
technique in terms of the jump of the discrete displacement over the
faces of the triangulation under consideration; in the second class of
elements, we use $\boldsymbol{H}_0^1(\Omega; \mathbb{R}^n)$-$P_{k}$ to
approximate the displacement space for $1\leq k\leq n$, and adopt the
stabilization technique suggested by Brezzi, Fortin, and Marini. We
establish the discrete inf-sup conditions, and consequently present
the a priori error analysis for them.  The main ingredient for the
analysis is two special interpolation operators, which can be
constructed using a crucial $\boldsymbol{H}(\mathbf{div})$ bubble
function space of polynomials on each element. The feature of these
methods is the low number of global degrees of freedom in the lowest
order case. We present some numerical results to demonstrate the
theoretical estimates.