Solution Method

In block form equation equations 6.1 and 6.2 takes the form of a saddle point problem

$\displaystyle \left[ \begin{array}{cc} A & B^{*} \\ B & 0 \\ \end{array} \right...
...p \\ \end{array} \right] =\left[ \begin{array}{c} G \\ 0 \\ \end{array} \right]$ (94)

where $ A$ is coercive, self-adjoint linear operator in a suitable Hilbert space, $ B$ is the $ (-1) \cdot$ divergence operator and $ B^{*}$ is it adjoint operator (=gradient operator). For more details on the mathematics see references [3,4]. We use iterative techniques to solve this problem. To make sure that the incomressibilty condition holds with sufficient accuracy we check for

$\displaystyle \Vert v\hackscore{k,k}\Vert \hackscore \le \epsilon \Vert\sqrt{v\hackscore{j,k}v\hackscore{j,k}}\Vert$ (95)

where $ \epsilon$ is the desired relative accuracy and

$\displaystyle \Vert p\Vert^2= \int\hackscore{\Omega} p^2 \; dx$ (96)

defines the $ L^2$-norm. We use the Uzawa scheme to solve the problem.

In fact the first equation in 6.5 gives for a known pressure

$\displaystyle v=A^{-1}(G-B^{*}p)$ (97)

which is inserted into the second equation leading to

$\displaystyle S p = B A^{-1} G$ (98)

with the Schur complement $ S=BA^{-1}B^{*}$. This problem can be solved iteratively with the preconditioner $ \hat{S}$ defined as $ q=\hat{S}^{-1}p$ by solving

$\displaystyle \frac{1}{\eta}q = p$ (99)

see [8] for more details. Note that the residual for the current approximation $ p$ is given as

$\displaystyle r=B A^{-1} (G - B^* p) = Bv$ (100)

where $ v$ is given by 6.8.

If one uses the generalized minimal residual method (GMRES) the method is directly applied to the preconditioned system

$\displaystyle \hat{S}^{-1} S p = \hat{S}^{-1} B A^{-1} G$ (101)

We use the norm

$\displaystyle \Vert p\Vert\hackscore{GMRES} = \Vert\hat{S} p \Vert$ (102)

Notice that for the residual $ \hat{r}=\hat{S}^{-1} r$ one has

$\displaystyle \ $ (103)

If $ p^{0}$ provides an initial guess for the pressure we use 6.8 to get a first initial guess for the velocity $ v^{0}$ which we use to set an absolute tolerance $ ATOL =\epsilon \Vert\sqrt{v^{0}\hackscore{j,k}v^{0}\hackscore{j,k}}\Vert$. The GMRES is terminated when

$\displaystyle \Vert\hat{r}\Vert\hackscore{GMRES} \le ATOL$ (104)

Notice that $ \Vert\hat{r}\Vert\hackscore{GMRES}= \Vert r \Vert = \Vert Bv\Vert = \Vert v\hackscore{k,k}\Vert$ so we we can expect that the target stopping criterion 6.6 is fullfilled. However, if $ v$ is very different from the initial choice of $ v^{0}$ the value of $ ATOL$ is corrected and GMRES is restarted with a new tolerance. For time dependend problems this apprach works well as value for $ p$ form a previous time step provides a good initial guess.

Alternatively, as $ S$ is symmetric and positive definite one can apply the preconditioned conjugate gradient method (PCG) . PCG use the norm

$\displaystyle \Vert r\Vert\hackscore{PCG}^2 = \int\hackscore{\Omega} r \hat{S}^{-1}r \; dx = \int\hackscore{\Omega} \eta r^2 \; dx$ (105)

To take the extra factor $ \eta$ into consideration when checking the stopping criterion we use the following definition for $ ATOL$:

$\displaystyle ATOL = \epsilon \frac{\Vert\sqrt{v^{0}\hackscore{j,k}v^{0}\hacksc...
...\Vert v^{0}\hackscore{k,k}\Vert} \Vert v^{0}\hackscore{k,k}\Vert\hackscore{PCG}$ (106)

esys@esscc.uq.edu.au