On the role of the Helmholtz-Leray projector in the space discretization of the Navier-Stokes equations
This is a modal window.
The media could not be loaded, either because the server or network failed or because the format is not supported.
Formal Metadata
| Title |
| |
| Title of Series | ||
| Number of Parts | 20 | |
| Author | ||
| Contributors | ||
| License | CC Attribution 3.0 Germany: You are free to use, adapt and copy, distribute and transmit the work or content in adapted or unchanged form for any legal purpose as long as the work is attributed to the author in the manner specified by the author or licensor. | |
| Identifiers | 10.5446/35363 (DOI) | |
| Publisher | ||
| Release Date | ||
| Language | ||
| Production Place | Leipzig |
Content Metadata
| Subject Area | ||
| Genre | ||
| Abstract |
|
Leibniz MMS Days 20183 / 20
00:00
EquationStokes' theoremSpacetimeJames Waddell Alexander IIWeightNumerical analysisComputer animationLecture/ConferenceMeeting/Interview
00:20
ForceEquivalence relationModel theoryComputational fluid dynamicsStagnation pointPressureContent (media)Point (geometry)Sign (mathematics)Maß <Mathematik>MathematicianRight angleDiameterMultiplication signLetterpress printingPartition (number theory)Film editingComputabilityMereologyDivisor1 (number)Social classConfidence intervalLie groupEquationEquivalence relationParameter (computer programming)State of matterPhysical lawTheoryRadical (chemistry)Numerical analysisComputer animation
02:42
Model theoryRandwertproblemEquationStokes' theoremEstimationStability theoryModulformForceSocial classEquivalence relationNormed vector spaceVelocityGradientPressureMereologyVelocityKörper <Algebra>PressureForcing (mathematics)Social classGradientFunctional (mathematics)Object (grammar)Equivalence relationArithmetic meanFree groupMechanism designMultiplication signGoodness of fitPoint (geometry)Distribution (mathematics)Mass flow rateComplex (psychology)DivergenceIntegration by partsPhysicalismINTEGRALNormal (geometry)Numerical analysisDirected graphResultantLine (geometry)MereologyNominal numberMoment (mathematics)Statistical hypothesis testingComputer animation
06:05
Model theoryComputational fluid dynamicsConstraint (mathematics)DivergenceInfinityElement (mathematics)Directed graphForceSocial classEquivalence relationReflection (mathematics)State of matterSocial classEquivalence relationMultiplication signComputer animation
06:29
Model theoryFunctional (mathematics)Vector spaceGradientMereologyDivergenceFree groupHelmholtz decompositionVector fieldConservation lawPressureObject (grammar)Order (biology)Fundamental theorem of algebraSpacetimeSurfaceExtension (kinesiology)Integration by partsNumerical analysisCausalityProjective planeRight angleQuantumNatural numberTheory of relativityVapor pressureComputer animation
08:57
Model theoryCategory of beingProof theoryEinbettung <Mathematik>Equivalence relationGradientFundamental theorem of algebraFunctional (mathematics)Integration by partsMechanism designCategory of beingAlgebraic structurePosition operatorProjective planeComputer animation
09:42
Stokes' theoremModel theoryEstimatorNormal (geometry)SphereComputer animation
10:13
PressureStokes' theoremSolution setLimit (category theory)Model theoryFluid staticsGradientVelocityRandwertproblemTerm (mathematics)Mass flow ratePressureFrictionGravitationWater vaporLimit (category theory)SpacetimeFree groupForcing (mathematics)Fluid staticsDivergenceChemical equationTheoryPerfect group1 (number)Different (Kate Ryan album)Right angleGraph (mathematics)Film editingDependent and independent variablesCondition numberComputer animation
11:57
Point (geometry)Saddle pointConstraint (mathematics)DivergenceModel theoryStokes' theoremAxonometric projectionNormal (geometry)ApproximationPressurePoint (geometry)VelocityProjective planeSpacetimeClassical physicsFinite element methodDivergenceFunctional (mathematics)Körper <Algebra>Basis <Mathematik>MomentumComputer animation
12:58
Model theoryPoint (geometry)Saddle pointConstraint (mathematics)DivergenceStokes' theoremAxonometric projectionCondition numberInterpolationExistenceVelocityInsertion lossPressureCategory of beingStatistical hypothesis testingFunctional (mathematics)MultilaterationSeries (mathematics)TheoryInterpolationEstimatorStokes' theoremComputer animation
13:43
Model theoryFluid staticsPressureLimit (category theory)Stokes' theoremKörper <Algebra>SpacetimeApproximationFunctional (mathematics)Free groupGradientTerm (mathematics)Integration by partsPressureProjective planeQuantum stateComputer animation
15:01
Model theoryEstimationStandard errorClassical physicsConservation lawMassStokes' theoremConvex hullKörper <Algebra>VelocityApproximationClassical physicsStandard errorParameter (computer programming)Right angleFunctional (mathematics)Multiplication signModel theoryDivisorPressureVector fieldInsertion lossArchaeological field surveyForcing (mathematics)Numerical analysisGroup actionVotingSupremumConservation lawComputer animation
16:19
Model theoryEstimationStandard errorClassical physicsConservation lawMassStokes' theoremApproximationOrder (biology)Water vaporSpacetimeStandard errorMetrologieMereologyPressureMass flow rate2 (number)PolynomialOrdinary differential equationPopulation densityRight angleLinearizationSocial classValidity (statistics)MetreComputer animation
17:51
Model theoryForceEquivalence relationSpacetimePressureStokes' theoremNumerical analysisStandard errorDirected graphGradientVortex2 (number)PressureSpacetimeOrder (biology)VelocityStability theoryReal numberGradientObject (grammar)Stagnation pointMass flow rateRight angleGraph coloringDirection (geometry)AreaPoint (geometry)Computer animation
19:04
Model theoryUniqueness quantificationForceFluid staticsMass flow ratePressureVelocityKörper <Algebra>LinearizationWater vaporComputer animation
19:46
Vector potentialModel theoryRandwertproblemPressureVelocityStagnation pointOrder (biology)GradientStandard errorPolynomialForcing (mathematics)Point (geometry)Computer animation
20:46
Model theoryGradientEquationMomentumTerm (mathematics)Körper <Algebra>Point (geometry)Stagnation pointVector potentialFinite element methodFluidMassMaß <Mathematik>Distribution (mathematics)PressureStagnation pointCategory of beingComplex numberElement (mathematics)Mass flow rateBoussinesq-ApproximationResultantMultiplication signComputer animation
21:29
Model theoryLink (knot theory)MassHill differential equationOrder (biology)Numerical analysisApproximationForcing (mathematics)Equivalence relationWater vaporSocial classPerturbation theoryPhysicalismComputer animation
22:28
Computational fluid dynamicsModel theoryContinuous functionPoisson processGroup actionFlow separationTerm (mathematics)ModulformMassOrder (biology)Term (mathematics)ModulformVector fieldGoodness of fitCivil engineeringTable (information)Computer animation
23:12
Model theoryTerm (mathematics)PressureVector spaceContinuous functionGradientWaveHamiltonian (quantum mechanics)Order (biology)RootFunctional (mathematics)Discounts and allowancesResidual (numerical analysis)Computer animation
23:33
Model theoryFinite element methodFunction (mathematics)Vector spaceDivergencePressureEstimationStandard errorNeumann boundary conditionValuation using multiplesMathematicsPhysical lawSpacetimeRight angleEstimatorStandard errorTerm (mathematics)WaveHamiltonian (quantum mechanics)Multiplication signForcing (mathematics)Social classEquivalence relationConsistencyObject (grammar)Computer animation
24:22
Model theoryPressureVector potentialClassical physicsOrder (biology)Finite element methodLine (geometry)WaveGradientHamiltonian (quantum mechanics)Reynolds numberPressureNumerical analysisVelocityBoundary layerKörper <Algebra>1 (number)Term (mathematics)Classical physicsOrdinary differential equationProjective planeModel theoryMathematicsRankingOrder (biology)Group actionElement (mathematics)Insertion lossChainDiffuser (automotive)Theory of relativityComputer animation
25:39
Vector potentialModel theoryReynolds numberPressureSurface energyPhase transitionMass flow rateVector fieldTerm (mathematics)Classical physicsFluid staticsMultiplication signTheory of relativitySurfacePressurePhase transitionGradientMany-sorted logicComputer animation
26:07
Arithmetic meanValuation using multiplesModel theoryEquivalence relationComputational fluid dynamicsStandard errorPressureMaxima and minimaMassConservation lawMathematicsOpen setElectric currentVelocityReynolds numberStagnation pointPoint (geometry)GradientOpen setComputational fluid dynamicsGradientOrder (biology)Computer animation
26:40
Lecture/ConferenceComputer animation
Transcript: English(auto-generated)
00:00
Okay, I'd like to talk about a relatively big thing that I'd like to talk about, which I don't want to talk about anymore. But now, I'd like to talk about a new topic. First, I'd like to mention that one of the problems, the numerator of the incompressible equation,
00:22
is a problem of equivalence of the classical equation. I'm not sure if I'm clear on this, but I'm going to talk about it later. However, this topic is not a practical topic. It's important to note that this topic,
00:41
the Hamilton-Lurei project, is the biggest mathematical instrument in the theory of theory. It doesn't have a numerator. I think it was eliminated. I'd like to talk about this topic
01:00
and then I'd like to talk about a very important point here. I'd like to talk about the last year of radicalization in this topic. How can the Navier-Stokes be so inconsistent with the criticism that we're talking about? And then there's the big debate. The law says it doesn't work.
01:22
We have a lot of simulations to talk about. And then, I'd like to talk about things that don't work and don't work. And here comes a point for two years I went to Shanghai for a conference and Nicholas Gauger had a question about the fact
01:40
that the state is so distant. First of all, we spent a lot of time at Imperial College which, with sex, we talked about in the beginning. And Nicholas Gauger said that we can't decide if we're going to be able to talk about sex
02:01
approximately and if we're going to be able to talk about sex. I can't talk about sex. And then, of course, the answer is consistent. And you can't talk about sex unless you're talking about a formal law
02:21
and I can't talk about that. And the point is the most robust method and my argument is that the new method that allows us to talk about the law of the algorithm is not so reducible because I believe it can be a fascinating regime.
02:41
Let's talk about it. So, let's talk about the influence of sex. Let's talk about it now. Here we have one of the most complex problems of the formulary and the mechanism to understand what this problem is. This is so simple in English.
03:03
For example, my brain doesn't know how to talk about sex. The multi-physical. We'll talk about the importance of sex We'll talk about how the test function works. This is the way it works. Let's see how it works.
03:21
In this term, we'll talk about the new gradient formulary or F-formulary. We'll talk about that. The gradient formulary means that F-formulary or gradient formulary and this formulary
03:41
means that this formulary is the object of the formulary and this formulary means that the supreme formulary is the function of the formulary. This is the big point here. These are the examples. This is the way it works. This is the way that the structural system is.
04:01
If it's not a long-term work, then it depends on the one. This is the way it works. And the object of the formulary is the object of the formulary. This is the thing. This is F-formulary. Now, good.
04:20
We'll talk about that. Gradient formulary. Gradient formulary or female formulary. I should speak in English, right? Okay, sorry. Now, I plug in the gradient phi times W.
04:41
I make integration by parts phi minus divergence W and this is zero because W is divergence free. So this object here, if I plug in a gradient thing, it's zero. This means this object is a semi-norm
05:02
because some arbitrarily large L2 functions if they are a gradient of something, they can be zero. And we all know from, let's say, the back integral, whatever, if we have a semi-norm,
05:21
we have an equivalence classes of forces. This is very simple. And what is the equivalence classes of forces for Navier-Stokes, for the incompressible one? F is the same like F plus any gradient. You can have any force which is a gradient as large as you want.
05:41
It will not be seen by the velocity field. It will go in the pressure. The pressure is completely determined by the velocity. It's a by-product. It doesn't matter for the complexity of the flow. Not at all. The velocity is determined by
06:01
equivalence classes. This is the main message here. And in my opinion, this equivalence class is one of the main confusions in CFD. It's one of the most basic things in design, where the paper is, I would say, the
06:21
state of the understanding from 2015 reflected. Which we published last year. Okay. Now I want to go to the Helmholtz-Lury projector in order to understand what happens here. So, if you have an L2 function, we can decompose it in some gradient part.
06:43
And this W part. And W is a divergent free function from that space L to sigma. One of the most fundamental objects in Navier-Stokes. This is a space all L2 vector fields which are orthogonal to all gradients.
07:02
What is this? Make integration by parts and we will see these are only the functions from L2 which are divergent free. And this is the marvellous thing. An L2 function divergent free is really orthogonal to all gradient fields.
07:20
And in my numerics, I will use that in order to make in an easy way, pressure robust methods. Let's exploit that. This has not been done in the past. Okay. And the Helmholtz projector now in L2 first is nothing else than the divergent free part.
07:43
Fine. The Helmholtz LeRue projector, this is what I did not understand for many years I would say. This is just the easy way how to go to H-1 because I always argued with H-1 with L2 but u grad u is not in L2 of course.
08:00
It's only in L1.5. So people ask me about my conservation but the Helmholtz LeRue projector is a very natural concept. It's a concept for functional and the function is just restriction. You have a function f and you restrict it to the divergent free functions.
08:21
This you call the Helmholtz LeRue projector. And if my f is in L2 I restrict it to the divergent free functions. This is the L2 Helmholtz decomposition. This is orthogonal. Then we get now the back Helmholtz decomposition in L2.
08:41
So it's really an extension. For all vector features in L2 Helmholtz LeRue is the same like Helmholtz. But it works for also for let's say surface tension as in your example with the bubbles and so on. It works for all H-1 things. OK.
09:01
Works for L5. This is the lowest thing where it works. In 3D due to so-called embedding. Anyway. This fundamental structure property of the Helmholtz LeRue projector is just every gradient is zero. This is the
09:20
equivalence class. Mechanism comes via the Helmholtz LeRue projector. And it's so easy. Gradify V integration by parts and this is restricted to divergent free functions. It's zero. And this is practically violated in old practical Navier-Stokes Discretizations.
09:43
OK. And now we can go back to our norm here with a V star and we see this thing is nothing else than the H minus 1 norm of the Helmholtz LeRue projector.
10:01
If you want to give a concept, a name to the whole thing. So it's a very natural thing. And we get this estimated with this semi-norm. I repeat myself. OK. Now an easy example. Stokes. And I want to say we have three limits. I want to speak now
10:21
about discretizations. And I say the classical discretizations work for some cases. And for one case not. I have this easy Stokes problem with the right hand side. And I have three different limit regimes I would say. One regime for example where the gradient of the pressure
10:41
is zero. This means they two terms balance and the boundary is zero. The velocity. OK. So it's divergent free force because this guy is divergent free. In coupled problems you can have such a thing. Then you can have this guy here where the
11:01
this guy is zero and this guy is a gradient and is balanced by the gradient of the pressure. Hydrostatic case. The glass of water. OK. There is a force. Gravity. But no flow. And the second, the third regime where we have now non-homogeneous
11:21
boundary conditions and they the gradient of the pressure balances exactly the friction term. This is Hagen-Poiseuille flow. And from these three regimes you can respond to the whole solution space of that guy here. And the classical schemes work for
11:40
this guy here. Perfect. And for this guy. And not at all for this. In theory first. But of course you have done something. Because this easy thing you want to have. You want to reproduce USD worldwide. Now we discussed this classic classical thing, Babushka-Bretzi.
12:03
What they did, OK, that's the conforming case here. We have some finite element spaces for the velocity field, finite element for the pressure space. And we have this discretely divergent free functions. And the thing is these discretely divergent free functions are not divergent free.
12:21
This is the main point. And Bretzi, Babushka, they all said it's not relevant. Doesn't matter. Just do it. OK. What we have now we have some best approximation in this discrete space here. The H1 norm and some L2 pressure
12:41
best approximation. And we define this discrete divergence. This is important. This is the main concept in mixed methods. We don't try to have divergence free zero but the projection onto the pressure space. And this is here the important point. How the pressure comes into something which is obviously a velocity property
13:03
of the discretization. This is the confusion. How pressure has something to do with velocities. Babushka-Bretzi theory says now we have some photon interpolator and we get the nice thing is now
13:21
divergent free functions are an optimally way approximated by discrete divergent functions. And this means Babushka-Bretzi are partially right. You can from the three regimes in Stokes we can handle in a perfect way.
13:41
This estimate. However, now we go to the hydrostatic regime. Hydrostatic regime if you do that you look now at a gradient
14:01
we introduce now a discrete Hammers-Lury projector. And we say the discrete Hammers-Lury projector is just applying to a functional discrete divergent free function. Very easy. It's not done in textbooks usually. Nobody speaks about Hammers-Lury. I repeat myself here.
14:23
So easy but this is the definition. Now you make grad 5 v8. Integration by parts is possible but now this guy is not divergent free. And the only thing what you can do you can plug in something the best approximation on the pressure space because
14:42
they are zero by definition of the scheme. And you see now the Hammers-Lury projector of a gradient is not zero but is this term here. Depends on the best approximability of phi on your pressure field. On your pressure space.
15:02
And here you get the error estimate. You can say my algorithm classical methods will give me the best approximation what is possible on this given grid for the velocity field up to an error which is only pressure dependent. And you have this horrible one over mu factor.
15:21
Where I say this comes from long time investigation and it tells us now if I have arbitrary large forces, gradient fields my continuous model would give me the zero function. But this guy would be arbitrary large away, arbitrary far
15:41
away from my best approximation of the velocity field. This is not so nice, right. And this is a locking phenomenon. What is a locking phenomenon? It means we have a small parameter like nu and we get tremendous errors. So what did Brubushka and Bredze do? They replaced one locking
16:01
phenomenon, so called Poisson locking by the locking phenomenon of poor mass conservation. This is what they did. One locking by another one. And now the however. Why does it work?
16:21
Yes, it's very easy because you can improve the pressure space. If you have a glass of water, your pressure will be linear. If you go to a you can make this error by Taylor approximation to zero.
16:41
If you use a pressure robust first order method, which we can construct now, we can get it with a first order method. But with the Bernardi-Roch-Jelck Roussouv, which are not robust, you will get tremendous errors for the simplest solution of a glass of water.
17:00
And this is very easy here in our glass of water example. But the typical problem in metrology is if the glass of water is as high as Mount Everest, so no flow problem. If this guy is Mount Everest, the density is temperature dependent
17:21
and the glass of water will not give you a linear polynomial of the pressure but a sixth order polynomial. And therefore in metrology codes people have hours like one meter per second if they have a mountain where they wouldn't expect any problems with no flow situation.
17:40
The whole problem of well balanced schemes, sorry, one of two parts of well balanced schemes have their origin in the same problem. Anyway. Okay. Now, one locking by another and the third second locking by by high order pressure spaces.
18:01
And by the velocity space has also been high. At least as high as the pressure space. Now we go to a stagnation point flow. Flow from the left to the right. And my claim to these things is people think these problems are difficult because we have
18:22
dominant advection. And my claim is it's not dominant advection, it's a pressure what makes a problem difficult. Here we have a flow coming here and here in the stagnation point we have the force balance u plus gradient
18:41
pi is zero. The pressure gradient drives around the object and makes the pressure very difficult here. Here is no pressure gradient. The same color of the pressure. You don't need that. But here where this thing is interesting, the pressure drives
19:01
everything. Okay. Now some examples. Glass of water I explained to you already. Glass of water we have the problem. The velocity field is zero but the pressure is linear. And what I want to say here, because people have not realized for many years
19:22
I would say, for decades, is that the pressure is usually more difficult than the velocity. They never cared about that question. Only Hagen-Poiseau is the only example where the pressure is more easy. It's linear and the velocity field is quadratic in channel flows.
19:42
But if it's more interesting the pressure will be more difficult. And now we go to potential. For example, one kind of stagnation point is steady potential flow. You can show in a steady potential flow all these forces are
20:01
gradients. One easy example you see here. For example, u is the same like the gradient of the Bernoulli pressure. To pick an example, velocity is let's say a polynomial of order four maybe. Then the pressure is of order eight a polynomial. It's much
20:21
more difficult. And this is my three and six example from If you approximate the boundary with third order the velocity will be third order overall. But the pressure error will be order six. It's absolutely reasonable to use a
20:41
sixth order method in order to make this pressure error small. This is an example here. You can have many this is complex numbers, you can have many stagnation points. And they all have the properties. The pressure is much more difficult. And this is a very old
21:01
problem. Let's say an old paperback forte. He said that something is wrong with incompressibility. He said Borussia-Beretti was fulfilled but we got completely unsatisfied results even for very simple flow problems.
21:22
1989. And what he proposed was Taylor-Hood element in a Boussinesq flow. And you can look at the literature now. So the question that I had for many years many people had that in the past.
21:40
So Grescho, Fontaine they were for Taylor-Hood because they understood the same problem with a glass of water and said make high order then the problem will be solved. But the problem is the more complex your Navier-Stokes problem is higher Reynolds number, more multiphysics, the same problem of equivalence classes of forces will reappear. Only
22:02
if you stick to one simple benchmark then you can say for this problem third order is enough. But if you want to go further you will have the same problems. But you will not understand it because you have never reflected on the order of approximation in your scheme.
22:21
And you can write many papers and say high order is much better. It's fine. And I think hundreds of publications are on that topic but people have never tried to go deeper, to go to Reynolds-Leray. They wrote one paper they said there are some phenomenon
22:42
high order was good, okay fine. And this, in many algorithms now you cannot go higher with an order. You have for example your code with Taylor-Holt, you cannot go to third order. And then the people invented quadratic stabilization, they tried to transform the pressure, they tried to have different
23:02
forms of an only convection term which only differ by gradient field again and again. This is what the people did. Also structured grids help you. Okay, and the easiest thing which I had in mind three years ago was you can use HDIF conforming
23:21
L2 functions in order to solve the problem at the root. We can make a new scheme where the discrete Hamiltonian wave projector of a gradient is always zero. It's easy with Rabi-Ato-Ma BDM spaces where we only change the right hand side. There's a troublemaker. Nowhere
23:41
else in Stokes. And what we get is a new estimate without any locking phenomenon. Okay, we get this nice estimate and the first time we have now it in MOSCOMM, the first paper that we have a data term now from this idea, consistency error here, but then now
24:01
what we get the consistency error is measured in the Hamiltonian wave projector of the Laplacian term. The first time the right equivalence class of forces appears in a paper where we have a data term, because not F is the data in Navier-Stokes, but the Hamiltonian wave projector
24:20
of delta U. Navier-Stokes, the same problem. As I said, u grad u can be a gradient if we have a potential flow and we change only here the discrete Hamiltonian wave projector by this easy local thing. We can compare two schemes classical Navier-Stokes I think it was a
24:40
P2 bubble, sorry Banoie de Roger, first order. We change the Reynolds number the newest one is Reynolds number one. We make higher Reynolds numbers, it becomes very bad, the scheme, but the new scheme where I only change this term here is nearly as good as here for Stokes. Because this is interesting, many Navier-Stokes
25:02
problems they give you a Stokes velocity field. It's very easy, no boundary layers at high Reynolds numbers because the pressure becomes complicated. Dominant advection is the wrong concept.
25:22
You can have dominant advection in Navier-Stokes of course but high Reynolds number is not the same like dominant advection. Because you have the Hamiltonian straight project of u grad u which matters not u grad u and if you stabilize u grad u you are on the wrong side.
25:40
These are related problems classic hydrostatic flows, laminar high Reynolds time dependent potential flows classic geostropic all the same thing, gradient fields balancing some other terms people have immense problems with geostropic flows. Two phase with surface tension and electrolytes
26:01
large pressures as Wolfgang Doier found out and these are the outlook I would say most important thing f is the same as f plus grad phi big confusion in CFD you can now make low order schemes which are much more accurate in some physical
26:22
regimes and these are the things which are related in my opinion many open questions for example relevant schemes in the compressible case thank you very much