### Choice of goal functional for PDE system

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Hi all,

I'm a Fenics newbie, and I'm trying to solve a couple PDE system of two equations. I have written my code based on this tutorial:

https://fenicsproject.org/pub/tutorial/html/._ftut1010.html#ftut1:reactionsystem

I'm trying to use an adaptive solver for the PDE, and I'm not sure how to define the goal functional needed for the adaptive step. Because the functional needs to be linear, I guess any choice must be of the form

a * sol[0] + b * sol[1] * dx

but I don't know how to make a meaningful choice for a and b.

What I'm currently doing is using a and b as weights so that the most 'difficult' function to compute has a higher relevance in the goal functional (by this I mean that it's got sharper features).

Can you please tell me what is the correct way to define a goal functional for a PDE system, and its motivation?

Thank you in advance!
Are you using the built in adaptive solver in FEniCS? Or formulating your own?

I'm a Fenics newbie, and I'm trying to solve a couple PDE system of two equations. I have written my code based on this tutorial:

https://fenicsproject.org/pub/tutorial/html/._ftut1010.html#ftut1:reactionsystem

I'm trying to use an adaptive solver for the PDE, and I'm not sure how to define the goal functional needed for the adaptive step. Because the functional needs to be linear, I guess any choice must be of the form

a * sol[0] + b * sol[1] * dx

but I don't know how to make a meaningful choice for a and b.

What I'm currently doing is using a and b as weights so that the most 'difficult' function to compute has a higher relevance in the goal functional (by this I mean that it's got sharper features).

Can you please tell me what is the correct way to define a goal functional for a PDE system, and its motivation?

Thank you in advance!

Community: FEniCS Project

written
8 months ago by
Nate

### 2 Answers

2

Goal-oriented adaptivity targets the case where you indeed have a specific goal functional in mind. So, the goal functional is something that should target your application. If you do not have an objective functional, some other type of norm-based adaptivity may be more appropriate.

But, for the record, if you are using the automated goal-oriented adaptive solve, the goal-functional does not need to be linear. It just needs to be UFL-expressable.

Being here, this paper of yours is great:

https://arxiv.org/abs/1204.6643

I'm posting it because other people may find it helpful in similar situations

But, for the record, if you are using the automated goal-oriented adaptive solve, the goal-functional does not need to be linear. It just needs to be UFL-expressable.

https://arxiv.org/abs/1204.6643

I'm posting it because other people may find it helpful in similar situations

written
8 months ago by
scaramouche-00

0

Thanks a lot for your reply. Based on what you said, and because my PDE system describes two coupled fields, I will attempt using the action as goal functional, as expressed in UFL. Thanks!

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