Hello Jose, I wish I could help, but I second Wolfgang's question. Is your code available somewhere? I would be glad to take a look at it and compare the solutions for the same problems using different formulations. I would expect that if you fix the issue with boundary conditions (those described in the Guermond paper, that is the "pressure boundary layer") then you would recover exactly what you should get with traditional schemes using Taylor-Hood element (as Martin discussed).
On another note, I remember having a discussion about this with Timo Heister at the deal.II workshop in 2019. Maybe Timo has ideas on this? I know he is quite the expert on algorithms to solve the Stokes / Navier-Stokes equations (e.g. his paper on the grad-div scheme, etc.) Sorry for not being to help more. Best Bruno On Sunday, October 18, 2020 at 12:21:21 p.m. UTC-4 jose.a...@gmail.com wrote: > > Hello everyone, > > I am working on a parallel solver based upon the incremental rotational > pressure-scheme as seen in the overview paper > <https://www.sciencedirect.com/science/article/pii/S0045782505004640?via%3Dihub#!> > > of Guermond and implemented in step-35 > <https://www.dealii.org/current/doxygen/deal.II/step_35.html>. To verify > the code, the following problems were replicated > > - Numerical test proposed in section 3.7.2 of Guermond's paper. It is > based on the method of manufactured solutions. The domain is the unit > square and a circle of radius 0.5. Dirichlet boundary conditions are > imposed on the whole boundary for the velocity. The pressure field is > constrained by setting its mean value to zero. > > > - The Taylor-Green vortex. The domain is the square (0,1)^2. The > velocity and pressure field are constrained by periodic boundary > conditions. Furthermore, the pressure field is constrained by setting its > mean value to zero. > > In order to compare the numerical pressure solution to its analytical > counterpart, the mean value of the numerical pressure field is made to > match that of the analytical one at the end of each time step. > > The results for the Guermond's numerical tests for the square domain > > Velocity convergence table > > ============================================================================================== > level dt cells dofs hmax L2 > H1 Linfty > 7 1.00e-01 8572 132098 1.10e-02 8.557567e-03 - > 6.456461e-02 - 1.103922e-02 - > 7 2.50e-02 8572 132098 1.10e-02 5.354964e-04 -2.00 5.090698e-03 -1.83 > 6.822255e-04 -2.01 > 7 6.25e-03 8572 132098 1.10e-02 3.358435e-05 -2.00 4.221552e-04 -1.80 > 4.245914e-05 -2.00 > 7 1.56e-03 8572 132098 1.10e-02 2.105764e-06 -2.00 3.632448e-05 -1.77 > 2.656204e-06 -2.00 > > Pressure convergence table > > ============================================================================================== > level dt cells dofs hmax L2 > H1 Linfty > 7 1.00e-01 8572 16641 1.10e-02 5.543160e-03 - 2.687030e-02 > - 3.649990e-02 - > 7 2.50e-02 8572 16641 1.10e-02 3.394874e-04 -2.01 3.684615e-03 -1.43 > 4.209359e-03 -1.56 > 7 6.25e-03 8572 16641 1.10e-02 2.126561e-05 -2.00 2.738788e-03 -0.21 > 3.958393e-04 -1.71 > 7 1.56e-03 8572 16641 1.10e-02 3.049911e-06 -1.40 2.727803e-03 -0.00 > 2.751900e-05 -1.92 > > and for the circle domain > > Velocity convergence table > > ============================================================================================== > level dt cells dofs hmax L2 > H1 Linfty > 6 1.00e-01 10979 164354 1.34e-02 7.593191e-03 - 5.404824e-02 > - 9.465067e-03 - > 6 2.50e-02 10979 164354 1.34e-02 4.879026e-04 -1.98 4.298417e-03 -1.83 > 6.072627e-04 -1.98 > 6 6.25e-03 10979 164354 1.34e-02 3.069737e-05 -2.00 3.539538e-04 -1.80 > 3.808221e-05 -2.00 > 6 1.56e-03 10979 164354 1.34e-02 1.926069e-06 -2.00 3.267549e-05 -1.72 > 2.500277e-06 -1.96 > > Pressure convergence table > > ============================================================================================== > level dt cells dofs hmax L2 > H1 Linfty > 6 1.00e-01 10979 20609 1.34e-02 2.855089e-03 - 1.235457e-02 > - 7.053574e-03 - > 6 2.50e-02 10979 20609 1.34e-02 1.983995e-04 -1.92 2.829075e-03 -1.06 > 4.839667e-04 -1.93 > 6 6.25e-03 10979 20609 1.34e-02 1.308016e-05 -1.96 2.705119e-03 -0.03 > 3.305446e-05 -1.94 > 6 1.56e-03 10979 20609 1.34e-02 3.290324e-06 -1.00 2.704622e-03 -0.00 > 2.323934e-05 -0.25 > > The Fig. 4 of the paper presents the log-log plot of the infinity norm of > the pressure vs time step size using 8 time step sizes. On the square > domain it shows convergence order of 1.6 on the square domain. Whereas on > the circle domain it shows a convergence order of 2.0 up to a time step > size of 2e-3. The last time step shown in Fig. 4 hints the reaching of the > asymptotic value of the error. > > The results for the Taylor-Green vortex problem > > Velocity convergence table > > ============================================================================================== > level dt cells dofs hmax L2 > H1 Linfty > 8 1.00e-01 17980 526338 5.52e-03 3.057096e-02 - 4.354711e-01 > - 4.998748e-02 - > 8 5.00e-02 17980 526338 5.52e-03 9.313187e-03 -1.71 1.289422e-01 -1.76 > 1.550793e-02 -1.69 > 8 2.50e-02 17980 526338 5.52e-03 2.493096e-03 -1.90 3.395720e-02 -1.92 > 4.112861e-03 -1.91 > 8 1.25e-02 17980 526338 5.52e-03 6.411600e-04 -1.96 8.668379e-03 -1.97 > 1.052791e-03 -1.97 > 8 6.25e-03 17980 526338 5.52e-03 1.623733e-04 -1.98 2.190470e-03 -1.98 > 2.661297e-04 -1.98 > 8 3.12e-03 17980 526338 5.52e-03 4.084716e-05 -1.99 5.601806e-04 -1.97 > 6.694580e-05 -1.99 > > Pressure convergence table > > ============================================================================================== > level dt cells dofs hmax L2 > H1 Linfty > 8 1.00e-01 17980 66049 5.52e-03 1.582475e-02 - 2.265999e-01 > - 4.902018e-02 - > 8 5.00e-02 17980 66049 5.52e-03 4.698715e-03 -1.75 6.899043e-02 -1.72 > 1.415426e-02 -1.79 > 8 2.50e-02 17980 66049 5.52e-03 1.249606e-03 -1.91 2.664252e-02 -1.37 > 3.746148e-03 -1.92 > 8 1.25e-02 17980 66049 5.52e-03 3.211439e-04 -1.96 2.054385e-02 -0.38 > 9.905812e-04 -1.92 > 8 6.25e-03 17980 66049 5.52e-03 8.191443e-05 -1.97 2.004909e-02 -0.04 > 2.840482e-04 -1.80 > 8 3.12e-03 17980 66049 5.52e-03 2.280548e-05 -1.84 2.000950e-02 -0.00 > 1.051814e-04 -1.43 > > While the velocity looks alright the H1 norm of the pressure locks pretty > fast and to a high value, while the L2 and infinity norm indicate a decline > in the convergence rate. > > I performed then a spatial convergence test on both problems, which > delivered for the Guermond problem > > Velocity convergence table > > ============================================================================================== > level dt cells dofs hmax L2 > H1 Linfty > 5 2.00e-02 361 8450 4.42e-02 3.390456e-04 - > 4.005312e-03 - 4.264292e-04 - > 6 2.00e-02 1228 33282 2.21e-02 3.379001e-04 -0.00 3.928219e-03 > -0.03 4.266117e-04 0.00 > 7 2.00e-02 4495 132098 1.10e-02 3.375465e-04 -0.00 3.881829e-03 > -0.02 4.266296e-04 0.00 > 8 2.00e-02 17170 526338 5.52e-03 3.374516e-04 -0.00 3.866594e-03 -0.01 > 4.265951e-04 -0.00 > > Pressure convergence table > > ============================================================================================== > level dt cells dofs hmax L2 > H1 Linfty > 5 2.00e-02 361 1089 4.42e-02 3.021009e-04 - 1.117284e-02 > - 1.817554e-03 - > 6 2.00e-02 1228 4225 2.21e-02 2.991258e-04 -0.01 6.139539e-03 > -0.86 2.426530e-03 0.42 > 7 2.00e-02 4495 16641 1.10e-02 2.988485e-04 -0.00 4.104065e-03 -0.58 > 2.943609e-03 0.28 > 8 2.00e-02 17170 66049 5.52e-03 2.988060e-04 -0.00 3.494146e-03 -0.23 > 3.440934e-03 0.23 > > and for the Taylor-green vortex > > Velocity convergence table > > ============================================================================================== > level dt cells dofs hmax L2 > H1 Linfty > 5 1.00e-02 472 8450 4.42e-02 4.176710e-04 - > 2.838026e-02 - 7.339885e-04 - > 6 1.00e-02 1444 33282 2.21e-02 4.135497e-04 -0.01 6.707143e-03 > -2.08 6.833520e-04 -0.10 > 7 1.00e-02 4912 132098 1.10e-02 4.127579e-04 -0.00 5.601192e-03 > -0.26 6.776613e-04 -0.01 > 8 1.00e-02 17980 526338 5.52e-03 4.125038e-04 -0.00 5.569924e-03 -0.01 > 6.767872e-04 -0.00 > > Pressure convergence table > > ============================================================================================== > level dt cells dofs hmax L2 > H1 Linfty > 5 1.00e-02 472 1089 4.42e-02 6.878170e-04 - 1.609285e-01 > - 3.492529e-03 - > 6 1.00e-02 1444 4225 2.21e-02 2.630181e-04 -1.39 8.027648e-02 > -1.00 1.329719e-03 -1.39 > 7 1.00e-02 4912 16641 1.10e-02 2.105590e-04 -0.32 4.018553e-02 -1.00 > 7.874550e-04 -0.76 > 8 1.00e-02 17980 66049 5.52e-03 2.067516e-04 -0.03 2.024105e-02 -0.99 > 6.528251e-04 -0.27 > > I was expecting the error to lock everywhere to the asymptotic value of > the error set by the time step but was surprised to see that on the > Taylor-Green vortex the refinement level sets a convergence rate of 1 on > the pressure H1-norm and has a weaker influence on the L2 and infinity > pressure norms and the H1-norm of the velocity. The Guermond problem shows > a similar behaviour but milder. My question to anyone with experience on > projection schemes and the Taylor-Green vortex, is whether this behaviour > is problem dependent, is intrinsic of the pressure-correction scheme, both > or am I overseeing an implementation/coding error and this behaviour should > not occurr at all? > > Extra info: A Taylor-Hood element was implemented. The convergence rates > shown in the tables are the experimental order of convergence using the > maximal circumradius of the triangulation (hmax) as reference column. The > level column indicates the number of global refinements. Please ignore the > cells column as on some tables the number of cells were obtained using the > wrong deal.ii method (n_active_cells instead of n_global_active_cells). > > Cheers, > Jose > -- 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. To view this discussion on the web visit https://groups.google.com/d/msgid/dealii/4bf37dc5-18ce-4521-8cc7-76a2bd2ce3fcn%40googlegroups.com.
