2 research outputs found
Fast Auxiliary Space Preconditioner for Linear Elasticity in Mixed Form
A block diagonal preconditioner with the minimal residual method and a block
triangular preconditioner with the generalized minimal residual method are
developed for Hu-Zhang mixed finite element methods of linear elasticity. They
are based on a new stability result of the saddle point system in
mesh-dependent norms. The mesh-dependent norm for the stress corresponds to the
mass matrix which is easy to invert while the displacement it is spectral
equivalent to Schur complement. A fast auxiliary space preconditioner based on
the conforming linear element of the linear elasticity problem is then
designed for solving the Schur complement. For both diagonal and triangular
preconditioners, it is proved that the conditioning numbers of the
preconditioned systems are bounded above by a constant independent of both the
crucial Lam\'e constant and the mesh-size. Numerical examples are presented to
support theoretical results. As byproducts, a new stabilized low order mixed
finite element method is proposed and analyzed and superconvergence results of
Hu-Zhang element are obtained.Comment: 25 page
New Hybridized Mixed Methods for Linear Elasticity and Optimal Multilevel Solvers
In this paper, we present a family of new mixed finite element methods for
linear elasticity for both spatial dimensions , which yields a
conforming and strongly symmetric approximation for stress. Applying
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 . For the lower order case
, 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.Comment: 30 page