Stable finite elements for the mixed form of the elasticity equations

Question

The mixed form of the elasticity equations is to find the unique critical point of the Hellinger-Reissner functional

$$J(u, \sigma) = \int_\Omega\left\{\frac{1}{2}A\sigma : \sigma + u\cdot\left(\nabla\cdot\sigma - f\right)\right\}dx$$

where $u$ is a vector field, $\sigma$ is a symmetric tensor field. The inputs are the rank-4 compliance tensor $A$ (which has some symmetries) and the forcing vector $f$. What finite elements are stable for this problem and what are their pros and cons? Alternatively, what stabilized formulations work?

Answer 1

The Arnold-Winther element (ref. 1) was the first stable element for mixed elasticity not requiring a relaxation of the symmetry of the stress tensor. For the lowest-order element in 2D, the displacement space $V_h$ consists of piecewise linear vector fields with no inter-element continuity requirements, which has dimension 6. The stress space $\Sigma_h$ consists of all piecewise cubic symmetric tensors such that the normal components are continuous across edges and all components are continuous at the vertices, which has dimension 24.

The extension of the Arnold-Winther element to 3D was published in 2008 (ref. 2). The lowest-order element is a tetrahedron that has piecewise linear displacements that are discontinuous between elements; this space ($V_h$) has dimension 12. The stress space $\Sigma_h$ contains the full space of quadratic polynomial on each element, augmented by divergence-free polynomials of degrees 3 and 4, with dimension 162.

References

1. Arnold, D. N., & Winther, R. (2002). Mixed finite elements for elasticity. Numerische Mathematik, 92(3), 401-419.

2. Arnold, D., Awanou, G., & Winther, R. (2008). Finite elements for symmetric tensors in three dimensions. Mathematics of Computation, 77(263), 1229-1251.

Answer 2

Brezzi, Fortin, and Marini showed that a version of the MINI element is stable for 2D elasticity. The stress and velocity spaces are $\Sigma_h = (\mathscr{L}_1^1 + \mathscr{B}_3)_s^{2\times 2}$, $V_h = (\mathscr{L}_1^1)^2$ where $\mathscr{L}_p^k$ is the space of polynomials of degree $p$ with $k$ continuous derivatives across triangle boundaries and $\mathscr{B}$ is the space of bubble functions, in this case cubic on each triangle. (The subscript ${}_s$ on the stress space denotes symmetric tensors.) In other words, the stress components are symmetrized, continuous piecewise linears, enriched with cubic bubbles, while the displacement components are continuous piecewise linears.

Unlike the $H(\text{div})$ elements, for example the Arnold-Winther element, the MINI element is continuous across triangle boundaries, which may be necessary for some applications.

References

Brezzi, F., Fortin, M., & Marini, L. D. (1993). Mixed finite element methods with continuous stresses. Mathematical Models and Methods in Applied Sciences, 3(02), 275-287.

Answer 3

In the PEERS (Plane Elasticity Element with Reduced Symmetry) element each row of the stress tensor is approximated by the lowest degree Raviart-Thomas element enriched by the curl of the cubic bubble element. This element has 4 stresses per vertex, 2 displacements per edge, and 1 Lagrange multiplier (rotation of displacement) per vertex. This gives a total of 21 degrees of freedom per element.

References

1. Arnold, D. N., Brezzi, F., & Douglas, J. (1984). PEERS: a new mixed finite element for plane elasticity. Japan Journal of Applied Mathematics, 1(2), 347-367.