A New Family of Efficient Conforming Mixed Finite Elements on Both Rectangular and Cuboid Meshes for Linear Elasticity in the Symmetric Formulation

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, the shear stress by the serendipity element of order k, and the displacement field by an enriched discontinuous vector-valued Pₖ₋₁ element. The degrees of freedom on each element of the lowest order element, which is of first order, are 10 plus 4. For three dimensions, the normal stress is approximated by an enriched Raviart--Thomas element of order k, 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 ≤ k-1 with respect to the rest variable, and the displacement field by an enriched discontinuous vector-valued Qₖ₋₁ element. The degrees of freedom on each element of the lowest order element, which is of first order, are 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 Pₖ₋₁ (resp., Qₖ₋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.