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Volume 40, Issue 2
Schur Complement Based Preconditioners for Twofold and Block Tridiagonal Saddle Point Problems

Mingchao Cai, Guoliang Ju & Jingzhi Li

DOI: 10.4208/cmr.2023-0051

Commun. Math. Res., 40 (2024), pp. 214-244.

Published online: 2024-05

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  • Abstract

In this paper, we consider using Schur complements to design preconditioners for twofold and block tridiagonal saddle point problems. One type of the preconditioners are based on the nested (or recursive) Schur complement, the other is based on an additive type Schur complement after permuting the original saddle point systems. We analyze different preconditioners incorporating the exact Schur complements. We show that some of them will lead to positively stable preconditioned systems if proper signs are selected in front of the Schur complements. These positive-stable preconditioners outperform other preconditioners if the Schur complements are further approximated inexactly. Numerical experiments for a 3-field formulation of the Biot model are provided to verify our predictions.

  • Keywords

Schur complement, block tridiagonal systems, positively stable preconditioners, Routh-Hurwitz stability criterion.

  • AMS Subject Headings

65F08, 65M22, 74F20

  • Copyright

COPYRIGHT: © Global Science Press

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@Article{CMR-40-214, author = {Cai , MingchaoJu , Guoliang and Li , Jingzhi}, title = {Schur Complement Based Preconditioners for Twofold and Block Tridiagonal Saddle Point Problems}, journal = {Communications in Mathematical Research }, year = {2024}, volume = {40}, number = {2}, pages = {214--244}, abstract = {

In this paper, we consider using Schur complements to design preconditioners for twofold and block tridiagonal saddle point problems. One type of the preconditioners are based on the nested (or recursive) Schur complement, the other is based on an additive type Schur complement after permuting the original saddle point systems. We analyze different preconditioners incorporating the exact Schur complements. We show that some of them will lead to positively stable preconditioned systems if proper signs are selected in front of the Schur complements. These positive-stable preconditioners outperform other preconditioners if the Schur complements are further approximated inexactly. Numerical experiments for a 3-field formulation of the Biot model are provided to verify our predictions.

}, issn = {2707-8523}, doi = {https://doi.org/10.4208/cmr.2023-0051}, url = {http://global-sci.org/intro/article_detail/cmr/23088.html} }
TY - JOUR T1 - Schur Complement Based Preconditioners for Twofold and Block Tridiagonal Saddle Point Problems AU - Cai , Mingchao AU - Ju , Guoliang AU - Li , Jingzhi JO - Communications in Mathematical Research VL - 2 SP - 214 EP - 244 PY - 2024 DA - 2024/05 SN - 40 DO - http://doi.org/10.4208/cmr.2023-0051 UR - https://global-sci.org/intro/article_detail/cmr/23088.html KW - Schur complement, block tridiagonal systems, positively stable preconditioners, Routh-Hurwitz stability criterion. AB -

In this paper, we consider using Schur complements to design preconditioners for twofold and block tridiagonal saddle point problems. One type of the preconditioners are based on the nested (or recursive) Schur complement, the other is based on an additive type Schur complement after permuting the original saddle point systems. We analyze different preconditioners incorporating the exact Schur complements. We show that some of them will lead to positively stable preconditioned systems if proper signs are selected in front of the Schur complements. These positive-stable preconditioners outperform other preconditioners if the Schur complements are further approximated inexactly. Numerical experiments for a 3-field formulation of the Biot model are provided to verify our predictions.

Cai , MingchaoJu , Guoliang and Li , Jingzhi. (2024). Schur Complement Based Preconditioners for Twofold and Block Tridiagonal Saddle Point Problems. Communications in Mathematical Research . 40 (2). 214-244. doi:10.4208/cmr.2023-0051
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