Ali,
As my first question (which is an elementary math level question) and
considering step-20 as an example, I was wondering how two equations
(bilinear forms) containing two unknowns can be summed up to a single
equation? While for example in the case of a system of two linear equations it
is not allowed. (my guess: bilinear forms are still in integral form)
I noticed this lack of understanding when I was trying to implement a vector
valued problem. It is of nonlinear type and composed of two equations. I
linearized and discretized equations separately. Now I have a 2*2 matrix
(consider A, B | A' , B') multiplied by {del.u ,del. v} (as unknowns) and on
the right hand side I have {f1, f2}.
Take the mixed Laplace equation as an example:
u + grad p = 0
-div u = f
If you multiply the first equation by a test function v and the second by a
test function q, then you will get the following two weak forms after
integration by parts (ignoring boundary conditions for a moment):
(v,u) - (div v, p) = 0 \forall v
-(q, div u) = (q,f) \forall q
We probably agree on this, right?
Now, we generally add these equations together and then have
(v,u) - (div v, p) - (q, div u) = (q,f) \forall v,q
If I understand your question right, then you are asking about why we can add
these equations together? That is because it is true *for all v,q*, and
consequently it is in particular true if we choose q=0. But if we choose q=0,
then the previous equation simplifies to
(v,u) - (div v, p) = 0 \forall v
On the other hand, if we had chosen v=0, then we will get that
-(q, div u) = (q,f) \forall q
In other words, the presence of the test functions and the statement "for all
test functions" allows us to recover the original equations from the summed-up
equation.
Does that make sense?
Best
Wolfgang
--
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Wolfgang Bangerth email: bange...@colostate.edu
www: http://www.math.colostate.edu/~bangerth/
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