Question on the operator matrix (system matrix) of Step-20

[Krishnakumar Gopalakrishnan]

Once again, apologies for asking questions which are more oriented towards vector calculus than the deal.II library/code itself.

In Step-20, under the introduction section on solvers, it says:

"If we sort our degrees of freedom so that all velocity come before all pressure variables, then we can subdivide the linear system Ax=b into the following blocks:"

(MBTB0)(UP)=(FG),

I am a bit confused about the origins of this block-matrix, particular the "B" term in the top-right block (and to a lesser extent to its transpose term in the bottom-left block). I shall explain more below.

The confusing aspects

Recalling that the above system matrix is the discretized version of the original (continuous) PDEs, this corresponds to the assembly of the bilinear form:

(v,K−1u)
−(div v,p)
−(q,div u)

• Now, the top-left block M of the matrix operator is clear; We are multiplying two vector shape functions which leads to the so-called "mass matrix".
• But the entry in the top-right block, B is not clear.  This is just a product of a scalar shape function and the negative divergence of a vector shape function.  How is this operator, the gradient?
• Yes, the bottom-left block is actually the same as #2 above, right? ie. the product of a scalar shape function and the negative divergence of a vector shape function?  Why is this entry indicated as just the -div operator?

Apologies for asking all these vector-calculus questions in this forum. 

Regards,

Krishna

[Wolfgang Bangerth]

On 2/1/20 8:25 AM, Krishnakumar Gopalakrishnan wrote:
> *__*

>
> Recalling that the above system matrix is the discretized version of the
> original (continuous) PDEs, this corresponds to the assembly of the bilinear form:
>
> (v,K−1u)
> −(divv,p)
> −(q,divu)
>

> * Now, the top-left block M of the matrix operator is clear; We are

> multiplying two vector shape functions which leads to the so-called "mass
> matrix".

> * But the entry in the top-right block, B is not clear.  This is just a

> product of a scalar shape function and the negative divergence of a vector

> shape function. *How is this operator, the gradient?*

Integrate it by parts to obtain -(div v,p) = +(v, grad p) :-)

Best
W.

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Wolfgang Bangerth email: bang...@colostate.edu
www: http://www.math.colostate.edu/~bangerth/