Dear Jean-Paul, Thanks for your suggestion. I'll try to test with these two approaches.
Regards, Lixing On Friday, January 8, 2021 at 4:36:44 AM UTC+8 Jean-Paul Pelteret wrote: > Hi Lixing, > > Another tutorial that might be of interest to you to look at is step-44 > <https://www.dealii.org/current/doxygen/deal.II/step_44.html>. In that > tutorial, two discontinuous fields are condensed out in one of two ways > (there's a switch to choose which method is applied). The first approach > <https://www.dealii.org/current/doxygen/deal.II/step_44.html#Solidassemble_sc> > > uses the "local condensation" technique, where one block of the global > block-system matrix is augmented when doing the condensation. The second > approach > <https://www.dealii.org/current/doxygen/deal.II/step_44.html#Solidsolve_linear_system> > > incorporates condensation into the solver strategy for the linear system by > adopting a (nested) global Schur complement; unlike the first technique, > this one does not change the values in the stored in the system matrix. In > either case, just one field of a three field problem is solved for, and the > others computed as a post-processing step. > > I think that the first approach might cover the details of one possible > way of doing what you list as point (2) "element-wise static condensation". > However, the second method (using LinearOperators) is, in my opinion, much > easier to implement (although, in the end, both approaches have their > advantages and disadvantages). > > I hope that this helps a little. > > Best, > Jean-Paul > > > On 07.01.21 08:11, Lixing Zhu wrote: > > Dear Wolgang, > > 1. I looked into the step-51 of the tutorial. It does illustrate a > paradigm of segregating local DOFs and global DOFs. If I utilize this > paradigm, the workflow would solve the local DOFs first (virtual node of > the bubble function), which is a block matrix since bubble support from > adjacent cells is irrelevant. Then I substitute the solution to the system > related to global DOFs (vertices of the standard Lagrange shape function), > namely, stabilizing the system. However, this would require certain FEM > space in the barycenter of the element (like a FE_Center, in the same > fashion as FE_Face), which seems not provided in deal.ii. > > 2. On the other hand, I am more inclined to use element-wise static > condensation (I guess this is somewhat not a rational decision). Is there > any example of how to eliminate the virtual DOFs of bubbles from the global > system once we have eliminated them locally? I guess this route can reduce > the time of implementation since I can at least utilize the FE_Q_Bubbles > with its linear realization. > > 3. The bubbles in FE_Q_Bubbles for a quadratic and higher-order element > are not the standard bubbles. Are there any specific references to these > bubbles? or how hard it is to modify the bubble function in FE_Q_Bubbles > for my own choice? > > Thank you for your time in advance. > > Best > Lixing > > On Tuesday, January 5, 2021 at 1:21:56 PM UTC+8 Wolfgang Bangerth wrote: > >> >> Lixing, >> >> > I am trying to implement a stabiliazed weak form (e.g. >> advection-diffusion) >> > where the stabilization tensor is computed element-wise through a >> standard >> > bubble: \Pi(1-x_i^2). It seems that FE_Q_Bubbles should provides all I >> need, >> > but here are two things I am not quite clear about, >> > >> > 1. Is the definition of bubble function in FE_Q_Bubbles in [0,1] span >> or >> > [-1,1] span? >> >> Everything in deal.II is on the reference cell define as [0,1]^d. >> >> >> > Does it has the standard bubble shape (1 at element center and >> > vanishes at the edges)? The (2x_j-1)^{p-1} part is a little bit >> confusing to >> > me. The bubble function corresponds to linear element in FE_Q_Bubbles >> is the >> > standard bubble that I desire, but I am really confusing at the shape >> of this >> > expression at higher orders. >> >> For higher orders, you end up with multiple bubble functions, one for >> each >> j=0..dim-1. This wasn't quite clear from the documentation, and I'll >> submit a >> patch later for that. >> >> >> > 2. I tried to utilize this class (i.e., FE_Q_Bubbles). One issue is >> that it >> > takes the virtual nodes of the bubble function into account of the >> total DOFs, >> > which is not the way we prefer in the stabilization method. >> >> But the behavior is correct -- the class describes a finite element >> *space* >> and that space contains the bubble function. I think that what you are >> trying >> to do is to do static elimination of that degree of freedom right away, >> and >> that can be implemented as well but is not the philosophical view we >> generally >> take in deal.II if you select such an element. >> >> The question is what you want the finite element space to be (i) locally, >> on >> every cell, and (ii) globally. There are a number of tutorials that >> discuss >> these sorts of questions. I would encourage you to read through step-61 >> and >> step-51, for example. >> >> Best >> WB >> >> -- >> ------------------------------------------------------------------------ >> Wolfgang Bangerth email: [email protected] >> 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 [email protected]. > To view this discussion on the web visit > https://groups.google.com/d/msgid/dealii/ba1bb5e4-4344-4f19-b630-8dee8d86dc3dn%40googlegroups.com > > <https://groups.google.com/d/msgid/dealii/ba1bb5e4-4344-4f19-b630-8dee8d86dc3dn%40googlegroups.com?utm_medium=email&utm_source=footer> > . > > > -- 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. 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