Abstract
Two-by-two block matrices of special form with square matrix blocks arise in important applications, such as in optimal control of partial differential equations and in high order time integration methods.
Two solution methods involving very efficient preconditioned matrices, one based on a Schur complement reduction of the given system and one based on a transformation matrix with a perturbation of one of the given matrix blocks are presented. The first method involves an additional inner solution with the pivot matrix block but gives a very tight condition number bound when applied for a time integration method. The second method does not involve this matrix block but only inner solutions with a linear combination of the pivot block and the off-diagonal matrix blocks.
Both the methods give small condition number bounds that hold uniformly in all parameters involved in the problem, i.e. are fully robust. The paper presents shorter proofs, extended and new results compared to earlier publications.
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This paper is dedicated to the memory of Ivo Marek, who recently unexpectedly deceased. The author is particularly thankful for his long lasting friendship with Ivo Marek, which also resulted in establishing important contacts with other excellent numerical analysts in the Czech Republic.
This work was supported by The Ministry of Education, Youth and Sports of the Czech Republic from the National Programme of Sustainability (NPU II), project “IT4Innovations excellence in science—LQ1602”.
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Axelsson, O. Preconditioning of two-by-two block matrix systems with square matrix blocks, with applications. Appl Math 62, 537–559 (2017). https://doi.org/10.21136/AM.2017.0222-17
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DOI: https://doi.org/10.21136/AM.2017.0222-17