2026/07/19 by Xuehai Huang, Zheqian Tang
#math.NA #cs.NA
A lowest-order mixed finite element method is developed for the strain gradient elasticity (SGE) model in arbitrary dimensions. We take the physically meaningful third-order double stress tensor \boldsymbolΦ:=ι2grad\boldsymbolσ(\boldsymbolu)∈\mathbbS⊗ℝd as a primary variable and derive a distributional mixed formulation. The double stress is approximated by an \mathbbS⊗ℝd-valued extension of the lowest-order Raviart--Thomas element, while the displacement is approximated by the vector-valued linear Crouzeix--Raviart element. Thus, the method avoids both high-degree bubble enrichment and a Nitsche-type treatment of the higher-order boundary condition. We establish parameter-robust discrete stability, an optimal first-order error estimate for fixed parameters, and a complementary parameter-uniform O(ι1/2+h) error estimate with constants independent of both the size parameter ι and the Lamé coefficient λ. In the boundary-layer regime ι1/2\lesssim h, the latter retains a first-order convergence rate in h. We also develop a local quadratic post-processing and a hybridized formulation. Numerical experiments in two and three dimensions support the theoretical results.