New Hybridized Mixed Methods for Linear Elasticity and Optimal Multilevel Solvers
For computational mechanics researchers, this work provides a family of conforming mixed methods with strong symmetry and optimal convergence, along with efficient solvers, though the approach is incremental over existing mixed finite element techniques.
This paper introduces new mixed finite element methods for linear elasticity that achieve optimal convergence rates for stress and displacement when k ≥ n, and remain stable on special grids for lower-order cases. The methods are implemented via hybridization, and multilevel solvers for the 2D Schur complement are proven uniformly convergent with respect to grid size and Poisson's ratio.
In this paper, we present a family of new mixed finite element methods for linear elasticity for both spatial dimensions $n=2,3$, which yields a conforming and strongly symmetric approximation for stress. Applying $\mathcal{P}_{k+1}-\mathcal{P}_k$ as the local approximation for the stress and displacement, the mixed methods achieve the optimal order of convergence for both the stress and displacement when $k \ge n$. For the lower order case $(n-2\le k<n)$, the stability and convergence still hold on some special grids. The proposed mixed methods are efficiently implemented by hybridization, which imposes the inter-element normal continuity of the stress by a Lagrange multiplier. Then, we develop and analyze multilevel solvers for the Schur complement of the hybridized system in the two dimensional case. Provided that no nearly singular vertex on the grids, the proposed solvers are proved to be uniformly convergent with respect to both the grid size and Poisson's ratio. Numerical experiments are provided to validate our theoretical results.