A Hybridized Staggered Discontinuous Galerkin--Mixed Finite Element Method for Strain Gradient Elasticity

Abstract

We propose a hybridized staggered discontinuous Galerkin--mixed finite element method for the strain gradient elasticity model, a fourth-order singularly perturbed problem governed by a material length scale parameter $\iota$ and the Lam\'e constants $\lambda$ and $\mu$. Introducing the total stress together with the scaled Cauchy and hyper stresses as auxiliary unknowns, we recast the model as a first-order system and discretize the displacement and the total stress by staggered discontinuous Galerkin spaces, the two scaled stresses by Raviart--Thomas pairs, with symmetry imposed strongly on all three stresses. The higher-order Dirichlet boundary condition enters the variational formulation naturally, so that the scheme avoids the numerical boundary layer that limits displacement-based methods. By establishing an inf-sup condition on the symmetric subspace and a discrete Korn inequality, we prove algebraic stability and optimal convergence, both uniform in $\iota$ and $\lambda$. We further develop a hybridized scheme in which local static condensation leaves only two multipliers on the mesh skeleton as globally coupled unknowns. Numerical experiments confirm the predicted convergence rates and the parameter robustness.