Re: solvers for elliptic equations with dominant lower order terms

[James Pringle]

For anyone else who finds this thread with google. A couple of points that
helped me get over the issues I was having:

   1. Of the non-trilinos solvers, scipy was most robust in solving
   problems where the advective terms locally dominated over diffusive terms.
   (I could not get trilinos to run).
   2. I found that decomposing the domains into squares was more robust
   than decomposing it into triangles. This may be because I was computing the
   coefficients of equations with the gradient and divergence operators in
   fipy.
   3. However, *BY FAR *the most important thing I had to get right was
   making sure the spatial variation in the coefficients was well resolved by
   the grid/mesh. All of the things you learned when writing your own codes
   remains true when using someone else's black box solver If your problem
   is not well resolved, your solver will do something weird or fail. In my
   case I had issues with steep topography in some small parts of the in the
   realistic oceanographic problem I was solving; this will be familiar to
   ocean modelers. I blame the glaciers that carved the Laurentian Channel for
   my problems...

To find problem areas, I found it useful to add a transient term to the
equation (making, essentially, a heat equation) and solving it in time with
a simplified boundary condition of 1 on the boundaries and starting with
zero in the interior. As the solution evolved, it became easy to spot
trouble areas, since the "true" solution should stay bounded between 0 and
1 and should approach a steady solution of  1 everywhere eventually (as
long as diffusion exists everywhere; if this is not true your problem may
be under-constrained and certainly will be hard to solve). Where the
solution starts becoming >>1 or << 0 are good places to start thinking
about... especially if those locations are right next to each other. It is
also reassuring if the solution to the transient problem converges to the
steady solution in time.

Jamie

(p.s. how to solve the above heat equation problem with a BC of eta=1 on
all boundaries at the long time limit? Note that 1 is a solution of the
original elliptic equation above (as is any constant), and that it meets
the boundary condition. The elliptic equation is the steady state of the
corresponding heat equation. Thus 1 is a solution to the problem at the
limit of long time.)

On Thu, Jun 1, 2017 at 8:34 PM, Guyer, Jonathan E. Dr. (Fed) <
jonathan.gu...@nist.gov> wrote:

> Unfortunately, James notified me offline of a breakage in PyTrilinos on
> Macs. I need to find time to debug.
>
> - Jon
>
> > On Jun 1, 2017, at 11:37 AM, Daniel Wheeler 
> wrote:
> >
> > Hi James,
> >
> > I don't have any hints for preconditioners, but using Trilinos's GMRES
> > and then systematically changing the GMRES parameters and
> > preconditioners might help you find one that works. I have done this
> > in the past, for example,
> >
> > https://urldefense.proofpoint.com/v2/url?u=https-3A__gist.
> github.com_wd15_c28ab796cb3d9781482b01fb67a7ec2d-23file-
> 2Dunsegregatedequation-2Dpy-2DL187=DwICAg=c6MrceVCY5m5A_KAUkrdoA=
> 7HJI3EH6RqKWf16fbYIYKw=GzwUdgPQ01r5uE6757k5b_5G60EzT_ebxic5VzxCPLw=
> ecDSV0HowRiX7RxqKQIM8k2pN6MWtkHeJVE0FsRUHKA=
> >
> > You can also get information about how well the solver is converging.
> >
> > Cheers,
> >
> > Daniel
> >
> > On Wed, May 31, 2017 at 3:31 PM, James Pringle  wrote:
> >>
> >> Dear all --
> >>
> >> I need to solve a second order PDE in 2D of the form
> >>
> >> A(x,y)*(eta_xx+eta_yy)+B(x,y)*eta_x+C(x,y)*eta_y = 0
> >>
> >> This is an advective diffusive balance, with B and C representing the
> >> advective velocities.
> >>
> >>
> >> Over much of the domain the low order terms dominate. For some
> geometries,
> >> the default solver I am using (SciPy LinearLUSolver) is having trouble
> find
> >> an answer (especially when the characteristics defined by B and C form
> >> loops.
> >>
> >> Does anyone have recommendations for how to choose a better solver? (I
> have
> >> blindly tried all the ones in SciPy, the ones besides LinearLUSolver do
> >> worse).
> >>
> >> And are there any hints on how to use preconditioners? There is
> discussion
> >> of preconditioners and Trilinnos, but I can't find hints on how one
> would
> >> choose to use them and when they might help.
> >>
> >> Pointers to the broader literature are welcome!
> >>
> >> Thank you,
> >> Jamie Pringle
> >>
> >> ___
> >> fipy mailing list
> >> fipy@nist.gov
> >> https://urldefense.proofpoint.com/v2/url?u=http-3A__www.
> ctcms.nist.gov_fipy=DwICAg=c6MrceVCY5m5A_KAUkrdoA=
> 7HJI3EH6RqKWf16fbYIYKw=GzwUdgPQ01r5uE6757k5b_5G60EzT_
> ebxic5VzxCPLw=FwlQdDjrMNL-fZozgCzuozW7l47NnT1W-2UfJvEgSsg=
> >>  [ NIST internal ONLY: https://urldefense.proofpoint.
> com/v2/url?u=https-3A__email.nist.gov_mailman_listinfo_
> fipy=DwICAg=c6MrceVCY5m5A_KAUkrdoA=7HJI3EH6RqKWf16fbYIYKw=
> 

Re: solvers for elliptic equations with dominant lower order terms

[Guyer, Jonathan E. Dr. (Fed)]

Unfortunately, James notified me offline of a breakage in PyTrilinos on Macs. I 
need to find time to debug.

- Jon

> On Jun 1, 2017, at 11:37 AM, Daniel Wheeler  wrote:
> 
> Hi James,
> 
> I don't have any hints for preconditioners, but using Trilinos's GMRES
> and then systematically changing the GMRES parameters and
> preconditioners might help you find one that works. I have done this
> in the past, for example,
> 
> 
> https://gist.github.com/wd15/c28ab796cb3d9781482b01fb67a7ec2d#file-unsegregatedequation-py-L187
> 
> You can also get information about how well the solver is converging.
> 
> Cheers,
> 
> Daniel
> 
> On Wed, May 31, 2017 at 3:31 PM, James Pringle  wrote:
>> 
>> Dear all --
>> 
>> I need to solve a second order PDE in 2D of the form
>> 
>> A(x,y)*(eta_xx+eta_yy)+B(x,y)*eta_x+C(x,y)*eta_y = 0
>> 
>> This is an advective diffusive balance, with B and C representing the
>> advective velocities.
>> 
>> 
>> Over much of the domain the low order terms dominate. For some geometries,
>> the default solver I am using (SciPy LinearLUSolver) is having trouble find
>> an answer (especially when the characteristics defined by B and C form
>> loops.
>> 
>> Does anyone have recommendations for how to choose a better solver? (I have
>> blindly tried all the ones in SciPy, the ones besides LinearLUSolver do
>> worse).
>> 
>> And are there any hints on how to use preconditioners? There is discussion
>> of preconditioners and Trilinnos, but I can't find hints on how one would
>> choose to use them and when they might help.
>> 
>> Pointers to the broader literature are welcome!
>> 
>> Thank you,
>> Jamie Pringle
>> 
>> ___
>> fipy mailing list
>> fipy@nist.gov
>> http://www.ctcms.nist.gov/fipy
>>  [ NIST internal ONLY: https://email.nist.gov/mailman/listinfo/fipy ]
>> 
> 
> 
> 
> -- 
> Daniel Wheeler
> ___
> fipy mailing list
> fipy@nist.gov
> http://www.ctcms.nist.gov/fipy
>  [ NIST internal ONLY: https://email.nist.gov/mailman/listinfo/fipy ]


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Re: solvers for elliptic equations with dominant lower order terms

[Sergio Manzetti]

James, can you perhaps send the FipY code for one of these equations, or a 
similar variant of it? 

Thanks! 
Sergio 


Sergio Manzetti 

[ http://www.fjordforsk.no/logo_hr2.jpg ] 

[ http://www.fjordforsk.no/ | Fjordforsk AS ] [ http://www.fjordforsk.no/ |   ] 
Midtun 
6894 Vangsnes 
Norge 
Org.nr. 911 659 654 
Tlf: +47 57695621 
[ http://www.oekolab.com/ | Økolab  ] | [ http://www.nanofact.no/ | Nanofactory 
 ] | [ http://www.aq-lab.no/ | AQ-Lab  ] | [ http://www.phap.no/ | FAP ] 



From: "James Pringle" <jprin...@unh.edu> 
To: "fipy" <fipy@nist.gov> 
Sent: Wednesday, May 31, 2017 10:50:11 PM 
Subject: Re: solvers for elliptic equations with dominant lower order terms 

I may have been unclear. This is a single PDE in the variable eta. A(x,y), 
B(x,y) and C(x,y) are coefficients of the PDE. They are indeed separate, and 
fipy does a nice job of solving this sort of equation in many limits. I am 
having problems when the lower order parts of the equation ( B(x,y)* 
eta_x+C(x,y)*eta_y) dominate the solution and the characteristics defined by B 
and C form closed loops. 

To understand this better, imagine the equation as a steady state fluid flow 
problem, with the A(x,y)*(eta_xx+eta_yy) representing diffusion (and to be 
correct, it should be written "Del . (A(x,y) Grad eta)"), and the low order 
terms representing advection, with B and C being the velocities doing the 
advection. When the diffusion is negligible, eta should be constant along the 
flow lines (characteristics) defined by B and C. But if those flow line form a 
closed loop, the diffusive term must determine the overall solution inside the 
loop... 

Thanks, 
Jamie 

On Wed, May 31, 2017 at 3:32 PM, Sergio Manzetti < [ 
mailto:sergio.manze...@fjordforsk.no | sergio.manze...@fjordforsk.no ] > wrote: 



James, are A(x,y), B(x,y) and C(x,y) three different variable functions for the 
PDE? 

FIPY can solve at least when these three are equal. If it solves when they 
differ, I am not sure. 

Sergio 


Sergio Manzetti 

[ 
https://urldefense.proofpoint.com/v2/url?u=http-3A__www.fjordforsk.no_logo-5Fhr2.jpg=DwMFaQ=c6MrceVCY5m5A_KAUkrdoA=7HJI3EH6RqKWf16fbYIYKw=INzNz0EhAfOz2J4uGWxrKjo-qBQYnUHjXOF9rSxEXTQ=U0xwV0Da6Bipu3Yzfcw9cPhJIwPF4QPA_zdG-Gms5uc=
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 | 
 ] 
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6894 Vangsnes 
Norge 
Org.nr. 911 659 654 
Tlf: [ tel:+47%2057%2069%2056%2021 | +47 57695621 ] 
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From: "James Pringle" < [ mailto:jprin...@unh.edu | jprin...@unh.edu ] > 
To: "fipy" < [ mailto:fipy@nist.gov | fipy@nist.gov ] > 
Sent: Wednesday, May 31, 2017 9:31:51 PM 
Subject: solvers for elliptic equations with dominant lower order terms 


Dear all -- 

I need to solve a second order PDE in 2D of the form 


BQ_BEGIN

A(x,y)*(eta_xx+eta_yy)+B(x,y)*eta_x+C(x,y)*eta_y = 0 



This is an advective diffusive balance, with B and C representing the advective 
velocities. 

BQ_BEGIN


BQ_END
Over much of the domain the low order terms dominate. For some geometries, the 
default solver I am using (SciPy LinearLUSolver) is having trouble find an 
answer (especially when the characteristics defined by B and C form loops . 

Does anyone have recommendations for how to choose a better solver? (I have 
blindly tried all the ones in SciPy, the ones besides LinearLUSolver do worse). 

And are there any hints on how to use preconditioners? There is discussion of 
preconditioners and Trilinnos, but I can't find hints on how one would choose 
to use them and when they might help. 

Pointers to the broader literature are welcome! 

Thank you, 
Jamie Pringle 

_

Re: solvers for elliptic equations with dominant lower order terms

[Daniel Wheeler]

Hi James,

I don't have any hints for preconditioners, but using Trilinos's GMRES
and then systematically changing the GMRES parameters and
preconditioners might help you find one that works. I have done this
in the past, for example,

 
https://gist.github.com/wd15/c28ab796cb3d9781482b01fb67a7ec2d#file-unsegregatedequation-py-L187

You can also get information about how well the solver is converging.

Cheers,

Daniel

On Wed, May 31, 2017 at 3:31 PM, James Pringle  wrote:
>
> Dear all --
>
> I need to solve a second order PDE in 2D of the form
>
> A(x,y)*(eta_xx+eta_yy)+B(x,y)*eta_x+C(x,y)*eta_y = 0
>
> This is an advective diffusive balance, with B and C representing the
> advective velocities.
>
>
> Over much of the domain the low order terms dominate. For some geometries,
> the default solver I am using (SciPy LinearLUSolver) is having trouble find
> an answer (especially when the characteristics defined by B and C form
> loops.
>
> Does anyone have recommendations for how to choose a better solver? (I have
> blindly tried all the ones in SciPy, the ones besides LinearLUSolver do
> worse).
>
> And are there any hints on how to use preconditioners? There is discussion
> of preconditioners and Trilinnos, but I can't find hints on how one would
> choose to use them and when they might help.
>
> Pointers to the broader literature are welcome!
>
> Thank you,
> Jamie Pringle
>
> ___
> fipy mailing list
> fipy@nist.gov
> http://www.ctcms.nist.gov/fipy
>   [ NIST internal ONLY: https://email.nist.gov/mailman/listinfo/fipy ]
>



-- 
Daniel Wheeler
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Re: solvers for elliptic equations with dominant lower order terms

[James Pringle]

I may have been unclear. This is a single PDE in the variable eta. A(x,y),
B(x,y) and C(x,y) are coefficients of the PDE. They are indeed separate,
and fipy does a nice job of solving this sort of equation in many limits. I
am having problems when the lower order parts of the equation
(B(x,y)*eta_x+C(x,y)*eta_y)
dominate the solution and the characteristics defined by B and C form
closed loops.

To understand this better, imagine the equation as a steady state fluid
flow problem, with the A(x,y)*(eta_xx+eta_yy) representing diffusion (and
to be correct, it should be written "Del . (A(x,y) Grad eta)"), and the low
order terms representing advection, with B and C being the velocities doing
the advection. When the diffusion is negligible, eta should be constant
along the flow lines (characteristics) defined by B and C.  But if those
flow line form a closed loop, the diffusive term must determine the overall
solution inside the loop...

Thanks,
Jamie

On Wed, May 31, 2017 at 3:32 PM, Sergio Manzetti <
sergio.manze...@fjordforsk.no> wrote:

> James, are A(x,y), B(x,y) and C(x,y) three different variable functions
> for the PDE?
>
> FIPY can solve at least when these three are equal. If it solves when they
> differ, I am not sure.
>
> Sergio
>
>
> Sergio Manzetti
>
>
> <https://urldefense.proofpoint.com/v2/url?u=http-3A__www.fjordforsk.no_logo-5Fhr2.jpg=DwMFaQ=c6MrceVCY5m5A_KAUkrdoA=7HJI3EH6RqKWf16fbYIYKw=INzNz0EhAfOz2J4uGWxrKjo-qBQYnUHjXOF9rSxEXTQ=U0xwV0Da6Bipu3Yzfcw9cPhJIwPF4QPA_zdG-Gms5uc=>
>
> Fjordforsk AS
> <https://urldefense.proofpoint.com/v2/url?u=http-3A__www.fjordforsk.no=DwMFaQ=c6MrceVCY5m5A_KAUkrdoA=7HJI3EH6RqKWf16fbYIYKw=INzNz0EhAfOz2J4uGWxrKjo-qBQYnUHjXOF9rSxEXTQ=U6qdDuDFx7m_FBcXq8wrKgktDC12I3ar-kaEGfdQ4kA=>
>
> <https://urldefense.proofpoint.com/v2/url?u=http-3A__www.fjordforsk.no=DwMFaQ=c6MrceVCY5m5A_KAUkrdoA=7HJI3EH6RqKWf16fbYIYKw=INzNz0EhAfOz2J4uGWxrKjo-qBQYnUHjXOF9rSxEXTQ=U6qdDuDFx7m_FBcXq8wrKgktDC12I3ar-kaEGfdQ4kA=>
> Midtun
> 6894 Vangsnes
> Norge
> Org.nr. 911 659 654
> Tlf: +47 57695621 <+47%2057%2069%2056%2021>
> Økolab
> <https://urldefense.proofpoint.com/v2/url?u=http-3A__www.oekolab.com=DwMFaQ=c6MrceVCY5m5A_KAUkrdoA=7HJI3EH6RqKWf16fbYIYKw=INzNz0EhAfOz2J4uGWxrKjo-qBQYnUHjXOF9rSxEXTQ=K6jUoxb_DR3t0-8jNTHjwDpAu7N5bdMUKatMC-pun-U=>
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> |  AQ-Lab
> <https://urldefense.proofpoint.com/v2/url?u=http-3A__www.aq-2Dlab.no=DwMFaQ=c6MrceVCY5m5A_KAUkrdoA=7HJI3EH6RqKWf16fbYIYKw=INzNz0EhAfOz2J4uGWxrKjo-qBQYnUHjXOF9rSxEXTQ=i8G56fQKIsGv9dk8ev8ozkaCLIsd-LOEHrqpX9cY5yI=>
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>
>
> ------
> *From: *"James Pringle" <jprin...@unh.edu>
> *To: *"fipy" <fipy@nist.gov>
> *Sent: *Wednesday, May 31, 2017 9:31:51 PM
> *Subject: *solvers for elliptic equations with dominant lower order terms
>
>
> Dear all --
>
> I need to solve a second order PDE in 2D of the form
>
> A(x,y)*(eta_xx+eta_yy)+B(x,y)*eta_x+C(x,y)*eta_y = 0
>
> This is an advective diffusive balance, with B and C representing the
> advective velocities.
>
>
> Over much of the domain the low order terms dominate. For some geometries,
> the default solver I am using (SciPy LinearLUSolver) is having trouble
> find an answer (especially when the characteristics defined by B and C form
> loops.
>
> Does anyone have recommendations for how to choose a better solver? (I
> have blindly tried all the ones in SciPy, the ones besides LinearLUSolver
> do worse).
>
> And are there any hints on how to use preconditioners? There is discussion
> of preconditioners and Trilinnos, but I can't find hints on how one would
> choose to use them and when they might help.
>
> Pointers to the broader literature are welcome!
>
> Thank you,
> Jamie Pringle
>
> ___
> fipy mailing list
> fipy@nist.gov
> http://www.ctcms.nist.gov/fipy
>   [ NIST internal ONLY: https://email.nist.gov/mailman/listinfo/fipy ]
>
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Re: solvers for elliptic equations with dominant lower order terms

[Sergio Manzetti]

James, are A(x,y), B(x,y) and C(x,y) three different variable functions for the 
PDE? 

FIPY can solve at least when these three are equal. If it solves when they 
differ, I am not sure. 

Sergio 


Sergio Manzetti 

[ http://www.fjordforsk.no/logo_hr2.jpg ] 

[ http://www.fjordforsk.no/ | Fjordforsk AS ] [ http://www.fjordforsk.no/ |   ] 
Midtun 
6894 Vangsnes 
Norge 
Org.nr. 911 659 654 
Tlf: +47 57695621 
[ http://www.oekolab.com/ | Økolab  ] | [ http://www.nanofact.no/ | Nanofactory 
 ] | [ http://www.aq-lab.no/ | AQ-Lab  ] | [ http://www.phap.no/ | FAP ] 



From: "James Pringle" <jprin...@unh.edu> 
To: "fipy" <fipy@nist.gov> 
Sent: Wednesday, May 31, 2017 9:31:51 PM 
Subject: solvers for elliptic equations with dominant lower order terms 


Dear all -- 

I need to solve a second order PDE in 2D of the form 




A(x,y)*(eta_xx+eta_yy)+B(x,y)*eta_x+C(x,y)*eta_y = 0 



This is an advective diffusive balance, with B and C representing the advective 
velocities. 

BQ_BEGIN


BQ_END
Over much of the domain the low order terms dominate. For some geometries, the 
default solver I am using (SciPy LinearLUSolver) is having trouble find an 
answer (especially when the characteristics defined by B and C form loops . 

Does anyone have recommendations for how to choose a better solver? (I have 
blindly tried all the ones in SciPy, the ones besides LinearLUSolver do worse). 

And are there any hints on how to use preconditioners? There is discussion of 
preconditioners and Trilinnos, but I can't find hints on how one would choose 
to use them and when they might help. 

Pointers to the broader literature are welcome! 

Thank you, 
Jamie Pringle 

___ 
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solvers for elliptic equations with dominant lower order terms

[James Pringle]

Dear all --

I need to solve a second order PDE in 2D of the form

A(x,y)*(eta_xx+eta_yy)+B(x,y)*eta_x+C(x,y)*eta_y = 0

This is an advective diffusive balance, with B and C representing the
advective velocities.


Over much of the domain the low order terms dominate. For some geometries,
the default solver I am using (SciPy LinearLUSolver) is having trouble find
an answer (especially when the characteristics defined by B and C form loops
.

Does anyone have recommendations for how to choose a better solver? (I have
blindly tried all the ones in SciPy, the ones besides LinearLUSolver do
worse).

And are there any hints on how to use preconditioners? There is discussion
of preconditioners and Trilinnos, but I can't find hints on how one would
choose to use them and when they might help.

Pointers to the broader literature are welcome!

Thank you,
Jamie Pringle
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