# Solvers for the Coupled Linear Systems of Equations

• Volker John
Chapter
Part of the Springer Series in Computational Mathematics book series (SSCM, volume 51)

## Abstract

Remark 9.1 (Motivation) Many methods for the simulation of incompressible flow problems require the simulation of coupled linear problems for velocity and pressure of the form
$$\displaystyle{ \mathcal{A}\underline{x} = \left (\begin{array}{*{10}c} A& D\\ B &-C \end{array} \right )\left (\begin{array}{*{10}c} \underline{u}\\ \underline{p} \end{array} \right ) = \left (\begin{array}{*{10}c} \underline{f}\\ \underline{f_{ p}} \end{array} \right ) =\underline{ y}, }$$
with
$$\displaystyle\begin{array}{rcl} & & A \in \mathbb{R}^{dN_{v}\times dN_{v} },\ D \in \mathbb{R}^{dN_{v}\times N_{p} },\ B \in \mathbb{R}^{N_{p}\times dN_{v} },\ C \in \mathbb{R}^{N_{p}\times N_{p} }, {}\\ & & \underline{u},\underline{f} \in \mathbb{R}^{dN_{v} },\ \underline{p},\underline{f_{p}} \in \mathbb{R}^{N_{p} }, {}\\ \end{array}$$
such that
$$\displaystyle{\mathcal{A}\in \mathbb{R}^{(dN_{v}+N_{p})\times (dN_{v}+N_{p})},\quad \underline{x},\underline{y} \in \mathbb{R}^{dN_{v}+N_{p} }.}$$
If C = 0, then (9.1) is a linear saddle point problem.

## Keywords

Linear Coupled System Linear Saddle Point Problem Incompressible Flow Incomplete Factorization Iterative Solver
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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© Springer International Publishing Switzerland 2016

## Authors and Affiliations

• Volker John
• 1
• 2
1. 1.Weierstrass Institute for Applied Analysis and StochasticsBerlinGermany
2. 2.Fachbereich Mathematik und InformatikFreie Universität BerlinBerlinGermany