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On the role of the Helmholtz-Leray projector in the space discretization of the Navier-Stokes equations

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this it prevalence classes is 1 of the main confusion and see if the some of the most basic things and this time whether paper that is I I would save the world the state of the art of stemming from profound 415 reflect which he published in the last year what
now I want to go to the mostly project in order to understand what happens to if you have a true function we candy-coat decompose it in some quantum parts and Zuse W. Part and W. is a divergent 3 function from that space L to the my 1 of the most fundamental objects and Navier-Stokes stocks this is the space of all true for vector feud which are orthogonal to all good regions what is this make integration by parts to receive these are only the fonts and honor to which a divergence-free and this is examined the marvelous saying no to function divergencies is really orthogonal to all gradient few in my numerics I will use that in order to make in these vapor pressure robust methods to exploit that just has not been done in the past OK and he hammered project on node troll for us is nothing I than the divergent right that was the way project this is what I did not understand for many years I would say this is just the easy way how to go to H minus 1 because I always argued with its minus 1 Ebbers's out to but you went his not an a to of course it's only in l 1 comma decimal 5 or so people ask me about my work on the relation that but the hand was right projectors is very nature concept the a concept for funds know and to pop defines as just restriction of a function f and you're restricted to the divergent functions this due cause with the right projector and if my half is no true I restricted to the divergent through functions so this is yet a true a composition this is orthogonal then we get this in we get now the Becsey at Hamas' composition 2 thirds reading extensions follow that defeats in to with new ways the same night hamlets but it works for also far let's say a surface tension as it in your example is a bubbles and so on it works for all X minus 1 thinks
what books for those 5 this is the most here at Brooks in Sweden to to stop off and bedding anyway and among the most
fundamental structural property of the array project is just every gradient this 0 this is a sacred rooms cost mechanism comes up why I was to protect and it's so easy gratified 5 really integration by parts of this guy and his is restricted to a deficit in the front and to work and this is what actually violated all practical eristics positions
OK and now we can get go back try 1 more because of the star and receive this thing is nothing else than the H minus 1 norm of C hemisphere a projector if you want to give a concept and name to the horse it's a very natural sick the does this estimator resistance my mom I repeat myself OK now an easy example
Stokes and I want to say yes to read limits I want to speak Knoblach discretization and as I say the president is conditioned verb for some cases and for 1 case not I have this easy stocks problems the right hand side and I have ced-3 different limit regimes are attack ones with him for example by the gradient of the pressure is the world this means they've through transparent and the real the bound well with the allotted to diverges 3 floors because is guys the evidence 3 in cut problems of graphs such as then you can have this guy here rather about this guy is there's ro through the skies agreed to and an spread and for the written pressure hydrostatic case the glass of water or there's a false gravity but no flow and as the 2nd the 3rd a regime that we have now mom this on the conditions and they the great enough to pressure balance exact the friction this is hard Paul's orchestra then from C 3 so regimes account respond to how solution space of that cut here and the classical schemes work for this guy here perfect enforce guy and not at all for this in the way 1st but of course you have done because this is the saying I want to have you on reproduce your wrist you were right other Stokes's
concession classical saying bubbles cupboard with the data OK that's the conforming case here we have some fine as basis for the velocity field Parliament for the precious space and we have this this greedy divergently functions and the thing is this this could you don't prevent not diverges for you this is a point and Pepsi but were scared they all said it's not relevant doesn't matter trust what can what we have now we have some best approximation in these discrete spaces here the each 1 momentum but a true pressure best approximation and the defined as discrete divergent this is important as the main concept in mixed methods they don't try to have divergence-free 0 but the projection onto the push aspects and this is you the important point how the pressure comes into something that
is all procedural loss it to you property of the discretization this is a confusion how pressure has something to do with velocities the can
reversible abreast series as now we have some for time aboard later and he gets the nice thing is now divergent 3 functions out an optimum IeveI and approximated by discredit evidence sentence and this means a push correct CI partially right you can from this screen regimes in Stokes we can tool you can handling perfect way this test is estimated however we go to the
hydrostatic regime however static
regime and if you do with that look now at a great and introduce notice could protect when we say that this could were hit lamasery projectors just applying to a function or a disk which it divergent for function very easy it's not done in textbooks usually nobody speaks about is how much way a repeat myself so it is a bunch this is the definition Nobel you make correct 5 V eights integration by parts as possible but not so this guy's not divergent flee and on using what you can do you can plug in some saying the best approximation the precious but because they are us viewable by definition of the sky and 3 you see now the Hamas' ray project of regret it is not see but this this term here depends on the best approximate ability of phi or under pressure and a precious but it's and here you
get the le estimator you can say my I agree with some classical massive severe give me the best approximation what is possible on good for the riddles food up to now which is only pressure depend and have this whole bovine of an effect when I say this comes from long time investigation and it tells us now if have arbitrary large forces gradient fields my continuous model what it would give me the seal hunt but this guy would be arbitrary survey arbitrator 5 away from my best approximation of the loss of it it is not a nice approach and this is the blocking phenomenon Watterson locking phenomenon it means we have a small perimetre sup like new and they get tremendous awards the vote a were bush compared to the very place to find knocking phenomena so-called pop song rock buyers and looking for number of Paul mosques conservation this is what they did 1 minus 1 and knowledge the however why does it through it's very easy because you can
impose the precious space if you have a glass of water the pressure will be linear right if you go to Terre Haute to be you can make this I will buy Taylor approximation to you if you use a pressure robust 1st order method which we can construct now you can get it is a 1st order method but is about 90 was dead was the revelry of and not song or bust you forget tremendous hours for the simplest solution of a glass of water what can and this is where we use a year in our in class and want example but a typical problem metrology is is a glass of Florida's highest amount of arrests the from proper if the if this guy's Mt Everest the density is temperature dependent and the grass avoid having not give you a linear polynomials pressure but a 6 order but in all and therefore metrology coach people have I was like a 1 meter per 2nd if they have a a mountain that it wouldn't expect me any problems was no fluke situations the whole apartment off valid schemes or sorry Van of CPU parts of schemes have their origin the same pop anywhere
OK now 1 looking by another is and fight 2nd blocking
published Kapitsa by higher-order pressure spaces and by M sub stability is a real varsity space has also been high at least as hard as a perfect now we go to
our stagnation more point flow flow from the left were at and my claim to these things is people think these 2 problems of difficult because if dominant adventure Michael areas It's not dominant direction at a pressure what makes a problem difficult to have you ever flow coming here and the and the stagnation point with the force balance you worked you plus plus gradient pi is you will the pressure gradient wives around the object and X oppression very difficult here there's no pressure gradient the same color of suppression that you don't need that but he had that this thing is interesting the pressure tribes of research what can now some
examples glass of what I explained you where did last affordably have the problem of the velocity field is 0 but the pressure is linear and over what I want to say here because people have not realized for many years ever save a daycare decades is that the pressure is usually more difficult than the velocity that was never cared about that question already had for size the only example but the precious more easy it's linear and the velocity it is quadratic and channel flows but if it's more interesting pressure will be more difficult and now we go to potential for
exam 1 kind of Thurmond of stagnation point is steady potential flow we can show understanding potential flow all of these forces gradients von easy example you see here but look what you want is the same like the gradient of the brain or pressure to produce example velocities that's a polynomial of of all order 4 maybe then the pressures of order point known it's much more difficult and this is my 3 and 6 example phone has been such if approximated bound referred although the velocity view the I work at the 3rd order of all but the pressure were really order sects it's absolutely reasonable to use a 6th or the method in order to makes us pressure were small this is an example it can have
many uses this complex numbers in Kenneth many stagnation crops and the the properties of pressures much more difficult OK
and and this is a very awkward for so but let's say all paperback time he said that something is wrong was encompass abyss of accomplice ability he said well bush of with the was fulfilled but we got completely unsatisfied results even for very simple for problems in 1989 and what he proposed was care the hold element in bossiness flow and you can look at the
literature now pose the question what I had for many years many people had that's the past of kind so regression falter they've they've FIL hot because they understood the same problem a glass of water make high all then the problem would be solved of a promise to the more complex your numbers toast problem is higher Reynolds number more mighty physics the same problem of equivalence classes of forces will reappear only if you stick to von simple benchmark then it then you can say OK for this problem so it always enough but if you want you you want to go further you who have the same problems but it will not understand it because you have never reflected on the order of approximation sky and you can write many papers and say hi all as much better it's fine all could and now I think
hundreds of publications on that topic because people have never tried to go deep up to go to let Harmison they they load from table is said that some phenomenon high-order order was caught OK fine and this will in many a I'd was snow you cannot go I haven't you have for example court Teleporter canonical to sort order and then the good and bad could of civilization it try to transform to pressure they tried to get different have different forms of a correction term which only differ by a great and feud again and again this is what the people can also structured goods help you OK and the easiest
thing which I had in mind this truth for 3 years ago was you can URIs H
discount conforming as to who's functions in order to to solve the problem as if the root leave the can make and new scheme residues could Hansbrough predictor of a gradient is oversee world easy ways of get
remark DM spaces only change the right
inside the as a troublemaker nowhere else in stocks and what you get is the new estimated without any knocking phenomenon OK get this nice
estimate and the 1st time we have knowledge in must come a 1st paper that of dada from his adware consistency it will ya but the novelty gets a consistent is measured in had much do they project office to the present the 1st times the wide equivalence classes of forces appears in a paper review have adopted from because not s so it's a dot eristics but the was rape objective Oscar can you the
Navier-Stokes the same problem as I said you would you can be a gradient in potential flow and the change our own yeah the discrete hammered away protected by this the local saying you can of compared to a schemes classic another that civil wars it's I think it was a P. true bobbled he Montessori by models 1st order the chain still number Zonnya's 1 is 1 of them 1 you really make higher elements numbers would becomes very bad for scheme but the news came there are only changes Tramp'' is nearly as good as here for stocks because this is interesting many Navier-Stokes problems the diffuse stokesley loss the field it's very easy no bond where there's at high Wells numbers because the pressure becomes complicated but dominant advection is a wrong concept you can have dominant edition of stocks of course but high ranks number is not the same like dominant of actions because you have was really project of to quote which matters and not you what you and it's just bless you when you you're on the wrong side they're related problems closet
had a steady flow stamina high rentals of time-dependent attentive post-grad it's just sort of all the same saying all the gradient feuds balancing to other terms people have immense promise just traffic flows to face surface tension actor large pressures there's it was going to ask for not and these are the valid look I would
say so most important as sustainment f PostScript fine as the coaches and see if the can now make Lauterstein schemes which are much more accurate in some physical receives and these are the things which are related to my opinion when many open questions for example Robin and schemes in the compressible camps thank you wrote
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Metadaten

Formale Metadaten

Titel On the role of the Helmholtz-Leray projector in the space discretization of the Navier-Stokes equations
Serientitel The Leibniz "Mathematical Modeling and Simulation" (MMS) Days 2018
Autor Linke, Alexander
Mitwirkende Leibniz-Institut für Oberflächenmodifizierung e.V. (IOP)
Leibniz-Institut für Troposphärenforschung (TROPOS)
Lizenz CC-Namensnennung 3.0 Deutschland:
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DOI 10.5446/35363
Herausgeber Weierstraß-Institut für Angewandte Analysis und Stochastik (WIAS), Technische Informationsbibliothek (TIB)
Erscheinungsjahr 2018
Sprache Englisch
Produktionsort Leipzig

Inhaltliche Metadaten

Fachgebiet Informatik, Mathematik
Abstract Recent progress on pressure-robust mixed methods for the incompressible Navier-Stokes equations and the connection to the Helmholtz-Leray projector is presented. It is argued that and how pressure-robust methods allow to decrease the (formal) approximation order of the used discretizations. As applications, electrolytes and stagnation point flows are emphasized.

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