Liang,
sorry but i think that your problem is not still clear! (sure more clear statement of problem helps to more effective feedback from others) > u_t + div(a u) =0 so do you mean u is a scaler quantity and a is a known velocity filed, then, how is configuration of your variables, i.e., which FE element you are using, and which interpolation, using disconttinuous Galerkin (DG) or continuous galerkin (CG)? btw, if your numerical treatment be fully conseravice in discrite form (e.g. using a suitable DG), physically you need div a = 0 to preserve mass during advection, so if your velocity field is fixed during simulation it should be primary div-free and also your interpolation/restriction operators should keep div a = 0. That paper i mention in previous gives a good sight. Beisdes to this paper, Mikhail Shshkov did good job in this regard, that support arbitrary mesh as well, he has several publication on this topic, i do not exactly mention them, so simply check his publication here: http://cnls.lanl.gov/~shashkov/ this one seems to be relevant: http://cnls.lanl.gov/~shashkov/papers/div-free.pdf Also, i think that using CG method it is not easy to enforce div-free interpolation but it should be simple in the case of DG due to local nature of DG that convert this global constraint to some local (path-wise) constraints. of course in the case of CG, it should be possible to enforce div-free interpolation (maybe expensive) hope this helps RT On Wed, Oct 22, 2008 at 6:11 AM, Zhu Liang <[EMAIL PROTECTED]> wrote: > Hi Yaqi > > Thank you very much for your reply. Unfortunately, tutorial 28 can not help > me. > The equation is rather simple. My problem is how to preserve the mass > during the mesh > refinement. To be specified, > > if I get the solution u_{n-1} at time n-1 on an "old" mesh, say T_{n-1}, > and I adaptively > refine the mesh to a new mesh T_n from the numerical solution u_{n-1}. Now > I want to > project u_{n-1} to the new refined mesh such that the mass is preserved, > i.e., the integration > of u_{n-1} on mesh T_{n-1} equal the integration of its projection on mesh > T_n. > > How to define this projection is my question. Of course, there maybe other > better approach for > this problem. If you have other good idea to solve this 3-D transport > equation, please let me know. > > Thank you again for your help. > > Best, > Liang > > On Tue, Oct 21, 2008 at 1:13 PM, Yaqi Wang <[EMAIL PROTECTED]> wrote: > >> Hi Zhu, >> >> I guess you are dealing with a term in your equation which composes of >> variables defined on a different mesh from the mesh of the equation. If this >> is true, you may want to take a quick look on tutorial 28. >> >> I hope this helps. >> >> Regards, >> yaqi >> >> On Tue, Oct 21, 2008 at 2:20 PM, Zhu Liang <[EMAIL PROTECTED]>wrote: >> >>> Hi all >>> >>> Sorry for the re-post. As Ruhollah suggested, I would like to repeat my >>> question in detail. >>> >>> The equation I faced is a simple 3-D transport equation with smooth >>> initial condition on a relatively large domain \Omega: >>> >>> u_t + div(a u) =0 >>> >>> In my analysis, u becomes singular as t goes to infinity. So I would like >>> to apply the adaptive finite element method (Crank-Nicolson scheme for time >>> discretization). My question is how to preserve the mass (\int_\Omega u >>> dx =const) during this approach. >>> >>> Basically, people apply the L^2- projection to transform numerical >>> solution u_{n-1} on an "old" mesh to a new refined mesh and make >>> the result the solution at time n-1. For this transport equation, I can >>> not use the L^2-projection as it violate the conservation law. >>> >>> Does anyone have some good ideas or suggestions for this case? >>> >>> Best, >>> Liang >>> >>> _______________________________________________ >>> >>> >> > > _______________________________________________ > >
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