Additive Inexact Block Triangular Preconditioners for Saddle Point Problems Arising in Meshfree Discretization of Piezoelectric Equations

Source abstract

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.