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Volume 12, Issue 2
Additive Inexact Block Triangular Preconditioners for Saddle Point Problems Arising in Meshfree Discretization of Piezoelectric Equations

Yang Cao, Qin-Qin Shen & Ying-Ting Chen

DOI: 10.4208/eajam.250921.120122

East Asian J. Appl. Math., 12 (2022), pp. 381-405.

Published online: 2022-02

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  • 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.

  • Keywords

Piezoelectric equation, element-free Galerkin method, saddle point problem, block triangular preconditioner, inexact Schur complement matrix.

  • AMS Subject Headings

65F10, 65F50

  • Copyright

COPYRIGHT: © Global Science Press

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@Article{EAJAM-12-381, author = {Cao , YangShen , Qin-Qin and Chen , Ying-Ting}, title = {Additive Inexact Block Triangular Preconditioners for Saddle Point Problems Arising in Meshfree Discretization of Piezoelectric Equations}, journal = {East Asian Journal on Applied Mathematics}, year = {2022}, volume = {12}, number = {2}, pages = {381--405}, 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.

}, issn = {2079-7370}, doi = {https://doi.org/10.4208/eajam.250921.120122 }, url = {http://global-sci.org/intro/article_detail/eajam/20260.html} }
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 -

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.

Yang Cao, Qin-Qin Shen & Ying-Ting Chen. (2022). Additive Inexact Block Triangular Preconditioners for Saddle Point Problems Arising in Meshfree Discretization of Piezoelectric Equations. East Asian Journal on Applied Mathematics. 12 (2). 381-405. doi:10.4208/eajam.250921.120122
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