Solvers for the Coupled Linear Systems of Equations

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 𝒜x̲=ADB−Cu̲p̲=f̲fp̲=y̲,$$\displaystyle{ \mathcal{A}\underline{x} = \left (\begin{array}{*{10}c} A& D\\ B &-C \end{array} \right )\left (\begin{array}{*{10}c} \underline{u}\\ \underline{p} \end{array} \right ) = \left (\begin{array}{*{10}c} \underline{f}\\ \underline{f_{ p}} \end{array} \right ) =\underline{ y}, }$$with A∈ℝdNvdNv,D∈ℝdNvNp,B∈ℝNpdNv,C∈ℝNpNp,u̲,f̲∈ℝdNv,p̲,fp̲∈ℝNp,$$\displaystyle\begin{array}{rcl} & & A \in \mathbb{R}^{dN_{v}\times dN_{v} },\ D \in \mathbb{R}^{dN_{v}\times N_{p} },\ B \in \mathbb{R}^{N_{p}\times dN_{v} },\ C \in \mathbb{R}^{N_{p}\times N_{p} }, {}\\ & & \underline{u},\underline{f} \in \mathbb{R}^{dN_{v} },\ \underline{p},\underline{f_{p}} \in \mathbb{R}^{N_{p} }, {}\\ \end{array}$$such that 𝒜∈ℝ(dNv+Np)(dNv+Np),x̲,y̲∈ℝdNv+Np.$$\displaystyle{\mathcal{A}\in \mathbb{R}^{(dN_{v}+N_{p})\times (dN_{v}+N_{p})},\quad \underline{x},\underline{y} \in \mathbb{R}^{dN_{v}+N_{p} }.}$$If C=0, then (9.1) is a linear saddle point problem.