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A new block triangular preconditioner for three-by-three block saddle-point problem

Jun Li, Xiangtuan Xiong (2024)

Applications of Mathematics

In this paper, to solve the three-by-three block saddle-point problem, a new block triangular (NBT) preconditioner is established, which can effectively avoid the solving difficulty that the coefficient matrices of linear subsystems are Schur complement matrices when the block preconditioner is applied to the Krylov subspace method. Theoretical analysis shows that the iteration method produced by the NBT preconditioner is unconditionally convergent. Besides, some spectral properties are also discussed....

A preconditioner for the FETI-DP method for mortar-type Crouzeix-Raviart element discretization

Chunmei Wang (2014)

Applications of Mathematics

In this paper, we consider mortar-type Crouzeix-Raviart element discretizations for second order elliptic problems with discontinuous coefficients. A preconditioner for the FETI-DP method is proposed. We prove that the condition number of the preconditioned operator is bounded by ( 1 + log ( H / h ) ) 2 , where H and h are mesh sizes. Finally, numerical tests are presented to verify the theoretical results.

Algebraic preconditioning for Biot-Barenblatt poroelastic systems

Radim Blaheta, Tomáš Luber (2017)

Applications of Mathematics

Poroelastic systems describe fluid flow through porous medium coupled with deformation of the porous matrix. In this paper, the deformation is described by linear elasticity, the fluid flow is modelled as Darcy flow. The main focus is on the Biot-Barenblatt model with double porosity/double permeability flow, which distinguishes flow in two regions considered as continua. The main goal is in proposing block diagonal preconditionings to systems arising from the discretization of the Biot-Barenblatt...

All-at-once preconditioning in PDE-constrained optimization

Tyrone Rees, Martin Stoll, Andy Wathen (2010)

Kybernetika

The optimization of functions subject to partial differential equations (PDE) plays an important role in many areas of science and industry. In this paper we introduce the basic concepts of PDE-constrained optimization and show how the all-at-once approach will lead to linear systems in saddle point form. We will discuss implementation details and different boundary conditions. We then show how these system can be solved efficiently and discuss methods and preconditioners also in the case when bound...

Applying approximate LU-factorizations as preconditioners in eight iterative methods for solving systems of linear algebraic equations

Zahari Zlatev, Krassimir Georgiev (2013)

Open Mathematics

Many problems arising in different fields of science and engineering can be reduced, by applying some appropriate discretization, either to a system of linear algebraic equations or to a sequence of such systems. The solution of a system of linear algebraic equations is very often the most time-consuming part of the computational process during the treatment of the original problem, because these systems can be very large (containing up to many millions of equations). It is, therefore, important...

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