Hi Vikram, Thank you for your reply. I am interested in solving PDEs with singular solutions. As you know, for PDEs without singularities (the solution is smooth enough), the conventional FEM can give very good results, and the convergence rate(error norm vs. element size) is optimal. However, when the solution involves singularities, for example, 1/r type solution, the standard FEM, I suppose, can only give either sub-optimal convergence rate or even diverge from the real solution. As a first step, I would like to know the performance of the standard FEM and adaptive FEM, such as their convergence rates, error bounds, and so on.
I will appreciate it if you can help. Thank you. Xujun On Mon, Oct 20, 2014 at 5:22 PM, Vikram Garg <[email protected]> wrote: > Hello Xujun, > For the mixed FEM formulation, the pressure usually > lies in the L2 function space, whereas the velocity lies in the H1 function > space. These function spaces dictate the norm to be used for measuring the > error in the variable. > > The patch recovery estimator assumes an H1 error norm, and would need to > be told that it should use L2 for the pressure variable. Can you tell us > what the code is for building your patch recovery error estimator, I can > then tell you the necessary modifications. > > Thanks. > > On Mon, Oct 20, 2014 at 5:16 PM, Xujun Zhao <[email protected]> wrote: > >> The true solution, u has 1/r singularity and pressure should have 1/r^2 >> singularity. So probably this is the case for singular problems, such as >> cracks. >> >> Yes, I limited the time of refinement in each element below 10 to make it >> stop. But the problem is that if the results do not converge, how can I >> trust my numerical results? is this the reason why 1/4 mid-point singular >> element is used in cracks? >> >> Xujun >> >> Xujun >> >> On Mon, Oct 20, 2014 at 5:07 PM, John Peterson <[email protected]> >> wrote: >> >> > On Mon, Oct 20, 2014 at 3:31 PM, Xujun Zhao <[email protected]> wrote: >> > > Hi all, >> > > >> > > For adaptive mesh refinement, an error estimator has to be used to >> > > approximately evaluate the errors in each element in order to >> determine >> > > which element is going to be refined. My question is which error >> > estimator >> > > should be used in the mixed FEM formulations, for example, Stokes >> > equation >> > > with u-p formulation. I read Bathe KJ's review paper: >> > > A posteriori error estimation techniques in practical finite element >> > > analysis, Computer & Structures 2005 p 235-265. >> > > in which they suggested another local error indicator. >> > > >> > > I did try kelly and patch recovery, and they are useful to locate the >> > > singular region and refine the mesh near the singularity. But the >> error >> > > norms become larger and larger with the refinement. It seems that the >> > > numerical solution is not convergent with mesh refinement. >> > >> > Is your true solution (especially for pressure) in H1? >> > >> > If not, like for the non-leaky lid-driven cavity, then Kelly will keep >> > giving you larger and larger error estimates. >> > >> > In practice, you can limit the amount of refinement in the singularity >> > by calling max_h_level() on the MeshRefinement object. >> > >> > -- >> > John >> > >> >> ------------------------------------------------------------------------------ >> Comprehensive Server Monitoring with Site24x7. >> Monitor 10 servers for $9/Month. >> Get alerted through email, SMS, voice calls or mobile push notifications. >> Take corrective actions from your mobile device. >> http://p.sf.net/sfu/Zoho >> _______________________________________________ >> Libmesh-users mailing list >> [email protected] >> https://lists.sourceforge.net/lists/listinfo/libmesh-users >> > > > > -- > Vikram Garg > Postdoctoral Associate > Center for Computational Engineering > Massachusetts Institute of Technology > http://web.mit.edu/vikramvg/www/ > > http://www.runforindia.org/runners/vikramg > ------------------------------------------------------------------------------ Comprehensive Server Monitoring with Site24x7. Monitor 10 servers for $9/Month. Get alerted through email, SMS, voice calls or mobile push notifications. Take corrective actions from your mobile device. http://p.sf.net/sfu/Zoho _______________________________________________ Libmesh-users mailing list [email protected] https://lists.sourceforge.net/lists/listinfo/libmesh-users
