Subject: [FEniCS-dev] boundary condition implementation

From: Gideon Simpson <[email-masked]>
Date: Mon, 7 Jan 2008 23:16:55 -0500

Something I discovered that I'm hoping someone will clarify is the  
following.

Let A be the matrix associated with a Finite Element formulation of  
Poisson's equation, as in the example in the demo directory.  If one  
exams the eigenvalues of A before the application of the BC's, there  
is a zero eigenvalue, while afterwards, there is no such troublemaker.

Thinking of A in terms of its associated weak form, I fully expected  
A to be a positive definite, symmetric matrix.  This is not the case.

-gideon

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From: "Garth N. Wells" <[email-masked]>
Date: Tue, 08 Jan 2008 08:17:57 +0000

Gideon Simpson wrote:
> exams the eigenvalues of A before the application of the BC's, there is 
> a zero eigenvalue, while afterwards, there is no such troublemaker. 
>

This is correct. What it's telling you is that you need to supply a 
Dirichlet boundary condition to ensure that the solution is unique.

> Thinking of A in terms of its associated weak form, I fully expected A 
> to be a positive definite, symmetric matrix.  This is not the case.
>
Inspecting of the form, I would expect a positive semi-definite matrix 
if you don't say anything about the value of functions at the boundary.

Garth

> 
> -gideon
> 
> 
> 
> ------------------------------------------------------------------------
> 
> _______________________________________________
> FEniCS-dev mailing list
> [email-masked]
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---

From: Gideon Simpson <[email-masked]>
Date: Tue, 8 Jan 2008 09:40:24 -0500

Ok, I see the error.  However, would you expect Poisson + zero  
Dirichlet data to no longer have a symmetric matrix?
-gideon

On Jan 8, 2008, at 3:17 AM, Garth N. Wells wrote:

>> following.  Let A be the matrix associated with a Finite Element  
>> formulation of Poisson's equation, as in the example in the demo  
>> directory.  If one exams the eigenvalues of A before the  
>> application of the BC's, there is a zero eigenvalue, while  
>> afterwards, there is no such troublemaker.
>
>> Thinking of A in terms of its associated weak form, I fully  
>> expected A to be a positive definite, symmetric matrix.  This is  
>> not the case.
>>
> Inspecting of the form, I would expect a positive semi-definite  
> matrix if you don't say anything about the value of functions at  
> the boundary.
>
> Garth
>
>> -gideon

---

From: "Garth N. Wells" <[email-masked]>
Date: Tue, 08 Jan 2008 14:45:01 +0000

Gideon Simpson wrote:
> Ok, I see the error.  However, would you expect Poisson + zero Dirichlet 
> data to no longer have a symmetric matrix?
>

The method DOLFIN uses to apply Dirichlet boundary conditions destroys 
symmetry. Other approaches which would preserve might eventually make it 
onto the TODO list . . .

Garth

>>> directory.  If one exams the eigenvalues of A before the application 
>>> of the BC's, there is a zero eigenvalue, while afterwards, there is 
>>> no such troublemaker. 
>>
>>> [email-masked] <mailto:[email-masked]>