A family of mixed finite elements for the elasticity problem
Numerische Mathematik
Mixed and hybrid finite element methods
Mixed and hybrid finite element methods
A Mixed Finite Element Method for Elasticity in Three Dimensions
Journal of Scientific Computing
Lower Order Rectangular Nonconforming Mixed Finite Elements for Plane Elasticity
SIAM Journal on Numerical Analysis
A Unified Analysis of Several Mixed Methods for Elasticity with Weak Stress Symmetry
Journal of Scientific Computing
Conforming Rectangular Mixed Finite Elements for Elasticity
Journal of Scientific Computing
Symmetric Nonconforming Mixed Finite Elements for Linear Elasticity
SIAM Journal on Numerical Analysis
Two Remarks on Rectangular Mixed Finite Elements for Elasticity
Journal of Scientific Computing
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We construct a family of lower-order rectangular conforming mixed finite elements, in any space dimension. In the method, the normal stress is approximated by quadratic polynomials $$\{1, x_{i}, x_{i}^{2}\}$${1,xi,xi2}, the shear stress by bilinear polynomials $$\{1, x_{i}, x_{j}, x_{i}x_{j}\}$${1,xi,xj,xixj}, and the displacement by linear polynomials $$\{1, x_{i} \}$${1,xi}. The number of total degrees of freedom (dof) per element is 10 plus 4 in 2D, and 21 plus 6 in 3D, while the previous record of least dof for conforming element is 17 plus 4 in 2D, and 72 plus 12 in 3D. The feature of this family of elements is, besides simplicity, that shape function spaces for both stress and displacement are independent of the spatial dimension $$n$$n. As a result of these choices, the theoretical analysis is independent of the spatial dimension as well. The well-posedness condition and the optimal a priori error estimate are proved. Numerical tests show the stability and effectiveness of these new elements.