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New Hybridized Mixed Methods for Linear Elasticity and Optimal Multilevel Solvers. (arXiv:1704.07540v1 [math.NA])
来源于:arXiv
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.
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