From d47f85023cb4ad91a1a4502518b18b5fa102a3d6 Mon Sep 17 00:00:00 2001 From: bangerth Date: Tue, 14 Apr 2009 21:47:12 +0000 Subject: [PATCH] Say something on the singularity of the matrix. git-svn-id: https://svn.dealii.org/trunk@18610 0785d39b-7218-0410-832d-ea1e28bc413d --- deal.II/examples/step-34/doc/intro.dox | 22 ++++++++++++++++++++-- 1 file changed, 20 insertions(+), 2 deletions(-) diff --git a/deal.II/examples/step-34/doc/intro.dox b/deal.II/examples/step-34/doc/intro.dox index ba305115b8..0d1da58479 100644 --- a/deal.II/examples/step-34/doc/intro.dox +++ b/deal.II/examples/step-34/doc/intro.dox @@ -527,14 +527,32 @@ With this choice of collocation points, the computation of the entries of the matrices $\mathbf{A}$, $\mathbf{N}$ and of the right hand side $\mathbf{b}$ requires the evaluation of singular integrals on the elements $K_i$ of the triangulation $\mathcal{T}_h$. - As usual in these cases, all integrations are performed on a reference simple domain, i.e., we assume that each element $K_i$ of $\mathcal{T}_h$ can be expressed as a linear (in two dimensions) or bi-linear (in three dimensions) transformation of the reference boundary element $\hat K := [0,1]^{n-1}$, and we perform the integrations after a change of variables from the real element $K_i$ to the reference -element $\hat K$. +element $\hat K$. + +Before discussing specifics of this integration in the +next section, let us point out that the matrix $\mathbf{A}+\mathbf{N}$ +is rank deficient. This is mostly easily seen by realizing that +$\mathbf{A}=-(\mathbf{N}\mathbf{e})\mathbf{e}^T$ where $\mathbf{e}$ is a +vector of all ones. Consequently, +$\mathbf{A}+\mathbf{N} = +\mathbf{N}(\mathbf{I}-\mathbf{e}\mathbf{e}^T)$. Even if $\mathbf{N}$ +has full rank, the resulting matrix has then clearly co-rank 1 with a +null space in the direction of $\mathbf{e}$, which is the space of +constant functions. + +As a consequence we will have to subtract the constant functions from +our numerical solution (which the linear solvers thankfully still +provides) to normalize it. On the other hand, the presence of +this kernel of the operator is irrelevant for our original purpose: +all we are interested in is the velocity, which equals the gradient of +$\phi$! +

Treating the singular integrals.

-- 2.39.5