TY - JOUR
T1 - Additive Inexact Block Triangular Preconditioners for Saddle Point Problems Arising in Meshfree Discretization of Piezoelectric Equations
AU - Cao , Yang
AU - Shen , Qin-Qin
AU - Chen , Ying-Ting
JO - East Asian Journal on Applied Mathematics
VL - 2
SP - 381
EP - 405
PY - 2022
DA - 2022/02
SN - 12
DO - http://doi.org/10.4208/eajam.250921.120122 
UR - https://global-sci.org/intro/article_detail/eajam/20260.html
KW - Piezoelectric equation, element-free Galerkin method, saddle point problem, block triangular preconditioner, inexact Schur complement matrix.
AB - <p style="text-align: justify;">Additive inexact block triangular preconditioners for discretized two-dimensional piezoelectric equations are proposed. In such preconditioners, the (1,1) leading
block is used as the block diagonal part of a discrete elasticity operator. The (2,2) block
is the approximation of the exact Schur complement matrix. It is additively assembled
by a small exact Schur complement matrix in each background cell. The proposed preconditioners are easy to construct and have sparse structure. It is proved that (1,1) and
(2,2) blocks of the preconditioners are spectrally equivalent to the (1,1) block of the
discretized piezoelectric equation and the exact Schur complement matrix, respectively.
Two numerical examples show that Krylov subspace iteration methods preconditioned
in this way, are fast convergent and the iteration steps do not depend on the degree of
freedom.</p>