From ce09de7df180cb25a643155f50fc671dbd3ee524 Mon Sep 17 00:00:00 2001 From: bangerth Date: Tue, 12 Aug 2008 02:58:30 +0000 Subject: [PATCH] Finish section on solvers for the Stokes part. git-svn-id: https://svn.dealii.org/trunk@16517 0785d39b-7218-0410-832d-ea1e28bc413d --- deal.II/examples/step-31/doc/intro.dox | 60 +++++++++++++++++++++++++- 1 file changed, 58 insertions(+), 2 deletions(-) diff --git a/deal.II/examples/step-31/doc/intro.dox b/deal.II/examples/step-31/doc/intro.dox index eda0be1fcf..7cd912b974 100644 --- a/deal.II/examples/step-31/doc/intro.dox +++ b/deal.II/examples/step-31/doc/intro.dox @@ -558,11 +558,67 @@ For this program, we would like to go a bit further. To this end, note that the symmetrized bilinear form on vector fields, $(\varepsilon {\mathbf v}_i, \eta \varepsilon ({\mathbf v}_j))$ is not too far away from the nonsymmetrized version, -$(\nabla {\mathbf v}_i, \eta \nabla ({\mathbf v}_j))$. The latter, +$(\nabla {\mathbf v}_i, \eta \nabla {\mathbf v}_j) += \sum_{k,l=1}^d + (\partial_k ({\mathbf v}_i)_l, \eta \partial_k ({\mathbf v}_j)_l) +$. The latter, however, has the advantage that the dim vector components of the test functions are not mixed, i.e. the resulting matrix is block-diagonal: one block for each vector component, and each of these -blocks is equal to the Laplace matrix for this vector component. +blocks is equal to the Laplace matrix for this vector component. So assuming +we order degrees of freedom in such a way that first all $x$-components of the +velocity are numbered, then the $y$-components, and then the $z$-components, +then the matrix $\hat A$ that is associated with this slightly different +bilinear form has the form +@f{eqnarray*} + \hat A = + \left(\begin{array}{ccc} + A_s & 0 & 0 \\ 0 & A_s & 0 \\ 0 & 0 & A_s + \end{array}\right) +@f} +where $A_s$ is a Laplace matrix of size equal to the number of shape functions +associated with each component of the vector-valued velocity. With this +matrix, we can now define our preconditioner for the velocity matrix $A$: +@f{eqnarray*} + \tilde A^{-1} = + \left(\begin{array}{ccc} + \tilde A_s^{-1} & 0 & 0 \\ + 0 & \tilde A_s^{-1} & 0 \\ + 0 & 0 & \tilde A_s^{-1} + \end{array}\right), +@f} +where $\tilde A_s^{-1}$ is a preconditioner for the Laplace matrix — +something where we know very well how to build good preconditioner! + +To sum this whole story up, we can observe: + +
Linear solvers for the temperature equation
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