A new family of efficient conforming mixed finite elements on both rectangular and cuboid meshes for linear elasticity in the symmetric formulation

Abstract: A new family of mixed finite elements is proposed for solving the classical Hellinger-Reissner mixed problem of the elasticity equations. For two dimensions, the normal stress of the matrix-valued stress field is approximated by an enriched Brezzi-Douglas-Fortin-Marini element of order $k$, and the shear stress by the serendipity element of order $k$, the displacement field by an enriched discontinuous vector-valued $P_{k-1}$ element. The degrees of freedom on each element of the lowest order element, which is of first order, is $10$ plus $4$. For three dimensions, the normal stress is approximated by an enriched Raviart-Thomas element of order $k$, and each component of the shear stress by a product space of the serendipity element space of two variables and the space of polynomials of degree $\leq k-1$ with respect to the rest variable, the displacement field by an enriched discontinuous vector-valued $Q_{k-1}$ element. The degrees of freedom on each element of the lowest order element, which is of first order, is $21$ plus $6$. A family of reduced elements is also proposed by dropping some interior bubble functions of the stress and employing the discontinuous vector-valued $P_{k-1}$ (resp. $Q_{k-1}$) element for the displacement field on each element. As a result the lowest order elements have $8$ plus $2$ and $18$ plus $3$ degrees of freedom on each element for two and three dimensions, respectively. 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.