Philip,
> The equations for finding these are exactly in step-20. But the symbols used
> are different.
Yes -- the beautiful universality of mathematics :-)
> 1) Is Raviet-Thomas elements specific for mixed Laplace equations? I
> understand that step-20 is a tutorial and for learning. Other then being a
> good learning opportunity, is there any specific reason why RT elements were
> chosen for step-20?
From the introduction of step-20:
"It is a well-known fact stated in almost every book on finite element theory
that if one chooses discrete finite element spaces for the approximation of
u,p inappropriately, then the resulting discrete saddle-point problem is
instable and the discrete solution will not converge to the exact solution.
To overcome this, a number of different finite element pairs for u,p have been
developed that lead to a stable discrete problem. One such pair is to use the
Raviart-Thomas spaces RT(k) for the velocity u and discontinuous elements of
class DQ(k) for the pressure p. For details about these spaces, we refer in
particular to the book on mixed finite element methods by Brezzi and Fortin,
but many other books on the theory of finite elements, for example the classic
book by Brenner and Scott, also state the relevant results."
> 1b) From another perspective, If I am solving for the mixed Laplace equation,
> do I need to setup the system to use the RT elements?
Yes. Or some of the other Hdiv elements.
> 2) The weak derivation for step-20 has Dirichlet boundary values incorporated
> into it. I believe the sentence is "Note how in this formulation, Dirichlet
> boundary values of the original problem are incorporated in the weak form." I
> am guessing that this is because of the p = g on d(omega) term. Later down
> the
> road, I might need to implement different B.C. such as a periodic,
> anti-periodic, or a mixed coefficient B.C. If this is the case, would I need
> to re-derive the weak form? Or can I utilize the one found in step-20? Or,
> would it be better for me to derive a more general weak form that can be used
> with any B.C.?
In general, whenever you integrate by parts, you get a boundary term and you
will have to think what to do with it. That's the place where you can apply
boundary values into the weak formulation. In the case of the mixed Laplace,
that just happens to be where Dirichlet boundary conditions go, just like with
the usual (non-mixed) formulation of the Laplace equation, the boundary term
allows you to take care of Neumann boundary conditions.
> 3) From what I have read so far, I get the impression that the R.T elements
> are also incorporated into the weak form. Basically this sentence here: "Both
> xhand whare from the space Xh=RT(k)×DQ(k), where RT(k)is itself a space"? is
> this thought correct? If so, if I wanted to change the element type later,
> would I need to re-derive the weak form?
No. You derive the weak formulation of the PDE, i.e., the continuous model. It
makes no reference to the (later step of) discretization.
> 4) The weak enforcement of the pressure boundary conditions, does this need
> to
> be factored in even if the B.C are not Dirichlet?
If you have Neumann boundary conditions, then you'll have to see what to do
with it in the context of the weak form you happen to have. I think I have a
whole video lecture on boundary conditions :-)
Best
W.
--
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Wolfgang Bangerth email: bange...@colostate.edu
www: http://www.math.colostate.edu/~bangerth/
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