Hello Wolfgang, Marthin, Bruno, Richard and Timo,
> 'm not entirely clear about what your question is. Are you seeing > convergence > rates that are too low or too large? It is not uncommon to have cases > where a > scheme converges too fast (the convergence rate is too large); this is > typically the case because the solution has a symmetry. > > Best > W. Apologies for not explaining myself clearer. I will try it again: >From my understanding, the lowest bound of the error on each norm is set either by the spatial or the temporal discretization. I was kind of expecting that the L2- and H1-Norms share a similar spatial and time dependence, i.e. that each field reaches its lowest bound simultaneously, and that they do so with a similar convergence rate evolution. Stated differently, they start with an order of convergence which remains constant for a given time step range. After reaching a small time step size, the convergence order tends to zero as the lowest bound of the error is reached. >From the tests' results I can see that H1-Norm of the pressure has a considerably stronger spatial dependence than the velocity, as it reaches its lowest bound while the velocity still has a constant convergence order. This behaviour is also seen in the L2- and Linfty-Norm but in a much more milder scale, as seen in the spatial convergence test. My question is if this stronger spatial dependency of the pressure is problem dependent or if it is intrinsic to the pressure-correction scheme. While I have not experimented in detail with the step-35 program, we have > done extensive studies on similar problems in > https://doi.org/10.1016/j.jcp.2017.09.031 (or > https://arxiv.org/abs/1706.09252 for an earlier preprint version of the > same manuscript) including the pressure correction scheme. While the > spatial discretization is DG where some of the issues are , the experience > from our experiments suggests that the pressure correction scheem should > not behave too differently from other time discretization schemes with > similar ingredients (say BDF-2 on the fully coupled scheme). In other > words, I would expect second order convergence in the pressure for > Taylor-Hood elements. I should note that there are some subtle issues with > boundary conditions in projection schemes, so I cannot exclude some hidden > problems with the step-35 implementation or the way you set up the > experiments. > > Best, > Martin > Thanks for the manuscript, there I notice that the results shown are those of the behaviour of the L2-Norm. My finite element implementation behaves similarly in the L2-Norm to the convergence rates in your paper (BFD2 leads to a 2nd order convergence on both velocity and, for the most part, on pressure). Did you also analyse the H1-Norm by any chance? There is where I see the stronger spatial dependency of the pressure. On another note, it caught my eye that you split the Neumann boundary conditions. I have not done tests with them yet, but what is the benefit of doing this or why is it necessary? Furthermore, my next step would be the DFG 2D-2 benchmark. There, you computed the traction force using the symmetric gradient instead of the normal gradient. While your formulation would be for me the correct formulation, as the stress tensor is so defined, on the benchmark papers they use the normal gradient. This has caused me some confusion, as to which formulation should I implement for the benchmark. 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). > I tried reformulating the question above. Hopefully it is clearer now. I will clean the code up and get back to you. The pressure-correction scheme I am using is the incremental rotational, which has the smallest error caused due to the boundary layer. Could the boundary layer in this case still cause such a strong influence? if the velocity error you have is low enough, the somewhat time-independent > PPE you solve given that velocity, > you might get high enough rates up to a certain point -> and that point > might lie below the error you see on those 3 levels. > So, try to go for smaller timesteps (keeping the Re the same) and use more > spatial refinement levels. > In general I would also recommend changing the Reynold's number a bit > around and see what happens - maybe it is an effect that is limited to low > Re? > > Anyways, having a bigger convergence rate than expected is a nice problem > to have, isn't it? ; ) > -I would not think it is caused by a bug given the other rates looking > just as expected! > > All the best, > Richard > Yes, this is on my todo list. Until now I have been working on my laptop but now I got access to a cluster so I can expand the scope of the tests. For the Guermond problem I had been using Re = 100 and for the Taylor-Green vortex Re = 5 and as you propose I should investigate of this effect is driven by the Reynolds number. Thanks for the suggestion! I am happy to give further comments, but -- like Wolfgang -- I don't > quite understand what the precise question is. That said: > > 1. With projection schemes you will need to be careful about pressure > boundary layers. A good starting point might be the Elman, Silvester, > Wathen book. > 2. Specific numerical test setups can be more or less sensitive to > this fact (size of pressure error vs velocity error, smoothness of > solutions in time, specific behavior on the boundary, ...) > > > -- > Timo Heister > Sorry for not being clearer. I reformulated the question at the beginning of the E-Mail. Thanks for the book suggestion, I will get on reading. The pressure-correction scheme I am using is the incremental rotational, for which the intrinsic boundary layer error is the smallest of all the schemes. Furthermore, for the Taylor-Green vortex I am using periodic boundary conditions on all boundaries which rules out the corner singularities which plague the pressure-correction scheme. The solution in itself is smooth, so I had not thought the error of the boundary layer could have had such an effect on the H1-Norm. Interestingly, while the Guermond problem has Dirichlet boundary conditions and a non-smooth boundary, it also shows a better behaviour on the H1-Norm. 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. 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