[aspect-devel] Solver for compostion equation and preconditioner for Stokes (Ying He)

[Wolfgang Bangerth]

Ying,
I still have this question of yours to answer :-(


>>> I am currently interested in developing the solver using Discontinuous
>>> Galerkin method for problems involving single compositional field,
>>> which has
>>> discontinuity but no Temperature dependency. The viscosity and
>>> density for the
>>> Stokes equation will be dependent only on the compositional field. I
>>> am a new
>>> person to “Aspect”.  All my work done right now is based on the
>>> modification
>>> of step-31 in deal.ii examples. However, when I use the same Data
>>> parameters
>>> (density, viscosity, initial condition etc.) and apply the FEM method
>>> to run
>>> both in Step-31 I modified and Aspect, the two results are very
>>> different.
>>>
>>> Below are my questions:
>>>
>>> 1, I didn’t find anything mentioned in the Aspect manual regarding the
>>> numerical scheme used for the compositional field.  I thought it
>>> might be the
>>> same as the temperature equation with default viscosity
>>> stabilization, however
>>> the results from aspect show large overshoot (around 15%) for the
>>> compositional field.
>>
>> Are you saying that if you set up a problem in such a way that the
>> temperature
>> and the compositional fields have the same initial conditions, that
>> you get
>> different results?
>
> I only used Aspect to run compositional fields problem (temperature
> independent), and used step-31 in deal.ii to run temperature problem
> with advection term only. Those two runs are applied with the same
> initial conditions and computational domain. I believe that the results
> should be essential the same, although I gave them two different names,
> as they are the solutions solved from the same system equations (stokes
> + time-dependent homogeneous advection).

step-31 and ASPECT are several years apart. There has been much 
development in ASPECT since we started with step-31, and I'm not 
surprised to hear that these programs do not produce corresponding 
results. What would concern me more is if the temperature and 
compositional solvers in ASPECT produce different results.

Among things that are different between step-31 and ASPECT are:
- The stabilization parameters for the advection scheme
- The fact that step-31 uses a semi-implicit advection scheme
   that treats the diffusion term implicitly and the advection
   term explicitly, whereas ASPECT is fully implicit.


> Sure. Two input files can be download at the following links.
> The first input is for the case with a ratio 1 of jump viscosity
> https://www.math.ucdavis.edu/~yinghe/in_gerya_deta1.prm
> The second input is for the case with a ratio 10 of jump viscosity
> https://www.math.ucdavis.edu/~yinghe/in_gerya_deta10.prm

These are ASPECT input files. But what do they show? Do you think that 
the solution they produce is wrong? Or just different from a 
corresponding setup for step-31?


>>> 2, in the aspect paper of M. Kronbichler, T. Heister, W. Bangerth in
>>> 2012, it
>>> gives a very detail explanation of the “poor but cheap”
>>> preconditioner used in
>>> Aspect. A better but expensive preconditoner will be applied if the
>>> cheap one
>>> failed for more than 30 iterations.  I am just wondering what kind of
>>> good but
>>> expensive precondioner used in Aspect, as for large variation viscosity
>>> problems, the cheap one is always not good enough.
>>
>> I think that's also explained in the paper, but the upshot is that the
>> poor
>> and cheap preconditioner uses
>>
>>    [X  B  ]
>>    [0 M^-1]
>>
>> as a preconditioner where X corresponds to one cycle of an algebraic
>> multigrid
>> on the velocity-velocity block A. The expensive but better
>> preconditioner uses
>> the same block structure as above, but X corresponds to A^{-1}
>> (solved with CG
>> and using the algebraic multigrid as a preconditioners).
>>
> Thanks for your clarification, although I thought that the word 'cheap'
> or 'expensive' also corresponds to the lower right schur complement
> matrix: 1/eta M^-1 --> cheap, and 'B A^{-1}B^T'--> expensive. So for
> aspect, is the lower right matrix always the matrix M^{-1} even for the
> jump viscosity case?

Correct. We never build the "true" Schur complement because that's 
computationally just not feasible.

Best
  W.


-- 
------------------------------------------------------------------------
Wolfgang Bangerth               email:            bangerth at math.tamu.edu
                                 www: http://www.math.tamu.edu/~bangerth/