We developed a new framework to design and analyze the mixed FEM for elasticity problems by establishing the following three main results: A structure of the discrete stress space: on simplicial grids, the discrete stress space can be  selected as the symmetric matrix-valued Lagrange element space, enriched  by a symmetric matrix-valued polynomial H(div) bubble function space  on each simplex; a  corresponding choice  applies to  product grids. Two basic algebraic results: (1) on each simplex, the symmetric matrices of rank one produced by the tensor products of the unit tangent vectors of the n(n+1)/2  edges of the simplex, form a basis of the space of the symmetric matrices; (2) on each simplex,  the divergence space of the above H(div) bubble function space is equal to  the orthogonal complement  space of the  rigid motion space with respect to the corresponding discrete displacement space (A similar result holds on a macroelement for the product grids). These define a two-step stability analysis which is new and different from the classic one in literature. As a result, on both simplicial and product grids, we were able to define the first families of both symmetric and optimal mixed elements with polynomial shape functions in any  space dimension. Furthermore, the discrete stress space has a simple basis which essentially consists of symmetric matrix-valued Lagrange element basis functions.
 

Jun Hu received his doctoral degree from Chinese Academy of Sciences in 2004 and is now a Professor of Mathematics at Peking University. His main research interest is in finite element methods of partial differential equations. He is the recipient of the first Youth Innovation prize of China Society for Computational Mathematics in 2015  and the National Excellent Doctoral Dissertation of PR China in 2006, and a former Alexander von Humboldt Research Fellow of Alexander von Humboldt Foundation of Germany in 2004. He serves as the editor of two journals: Advances in Applied Mathematics and Mechanics, and Computer and Mathematics with Applications.