Published

Fast Auxiliary Space Preconditioner for Linear Elasticity in Mixed Form

Long Chen, Jun Hu, and Xuehai Huang

Mathematics of Computation. Published electronically: November 9, 2017.

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ABSTRACT:

 A block diagonal preconditioner with the minimal residual
method and an approximate block factorization preconditioner with the
generalized minimal residual method are developed for Hu-Zhang mixed
finite element methods for linear elasticity. They are based on a new
stability result for 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 for the displacement it is
spectral equivalent to the Schur complement. A fast auxiliary space
preconditioner based on the $H^1$ 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 for the Hu-Zhang element are obtained.