A family of symmetric mixed finite elements for linear elasticity on tetrahedral grids

A family of stable mixed finite elements for the linear elasticity on tetrahedral grids are constructed, where the stress is approximated by symmetric H(\d)-$P_k$ polynomial tensors and the displacement is approximated by $C^{-1}$-$P_{k-1}$ polynomial vectors, for all $k\ge 4$. Numerical tests are provided.Comment: 11. arXiv admin note: substantial text overlap with arXiv:1406.745

Stabilized mixed finite element methods for linear elasticity on simplicial grids in Rn\mathbb{R}^{n}

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.Comment: 16 pages, 1 figur

New lower order mixed finite element methods for linear elasticity

New lower order $H(\textrm{div})$-conforming finite elements for symmetric tensors are constructed in arbitrary dimension. The space of shape functions is defined by enriching the symmetric quadratic polynomial space with the $(d+1)$-order normal-normal face bubble space. The reduced counterpart has only $d(d+1)^2$ degrees of freedom. In two dimensions, basis functions are explicitly given in terms of barycentric coordinates. Lower order conforming finite element elasticity complexes starting from the Bell element, are developed in two dimensions. These finite elements for symmetric tensors are applied to devise robust mixed finite element methods for the linear elasticity problem, which possess the uniform error estimates with respect to the Lam\'{e} coefficient $\lambda$, and superconvergence for the displacement. Numerical results are provided to verify the theoretical convergence rates.Comment: 23 pages, 2 figure