Hello Dr. Bangerth, Thank you for your response and clearing up those questions. I am continually going through steps 8, 20, and 21.
I ma currently exploring 2 directions, Jean-Paul linear elastic direction and the Mixed-Laplace direction. From what I can tell, modeling the anisotropic materials will not be a problem in deal.ii. Which is a nice thing. On Tuesday, November 6, 2018 at 9:17:45 AM UTC-5, Wolfgang Bangerth wrote: > > > 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. > > -- > ------------------------------------------------------------------------ > Wolfgang Bangerth email: bang...@colostate.edu > <javascript:> > www: http://www.math.colostate.edu/~bangerth/ > > -- The deal.II project is located at http://www.dealii.org/ For mailing list/forum options, see https://groups.google.com/d/forum/dealii?hl=en --- You received this message because you are subscribed to the Google Groups "deal.II User Group" group. To unsubscribe from this group and stop receiving emails from it, send an email to dealii+unsubscr...@googlegroups.com. For more options, visit https://groups.google.com/d/optout.
