Hey Xujun,
For a PDE with a singular solution, the conventional FEM
indeed gives sub-optimal convergence. This is because the optimal
convergence rates for the conventional FEM are obtained with a assumption
that the solution has a certain amount of smoothness, typically this
assumption is that it lies in some smooth Hilbert Space: H1 or H2.
A singular solution like the one you are describing will have lower
regularity in the region near the singularity. So if the solution is in H1
away from the singularity, it might be in between L2 and H1 in the singular
region. This causes a degeneration in the convergence rate for the
conventional FEM.
The total error in the approximation for the singular problem can then be
thought of as: total error = error_singularity + error_FEM.
An adaptive FEM should restore the optimal rate by refining elements near
the singularity and reducing the error_singularity to a very low level in
the preasymptotic region itself. Then we are left with only the error_FEM
in the asymptotic region which converges with the optimal rate.
I think it would be easier if you start with a non-singular version of your
problem, figure out the right settings for your error indicators and then
try the singular problem.
Thanks.
On Mon, Oct 20, 2014 at 5:49 PM, Xujun Zhao <[email protected]> wrote:
> Hi Vikram,
>
> Thank you for your reply. I am interested in solving PDEs with singular
> solutions. As you know, for PDEs without singularities (the solution is
> smooth enough), the conventional FEM can give very good results, and the
> convergence rate(error norm vs. element size) is optimal. However, when the
> solution involves singularities, for example, 1/r type solution, the
> standard FEM, I suppose, can only give either sub-optimal convergence rate
> or even diverge from the real solution. As a first step, I would like to
> know the performance of the standard FEM and adaptive FEM, such as their
> convergence rates, error bounds, and so on.
>
> I will appreciate it if you can help. Thank you.
>
> Xujun
>
> On Mon, Oct 20, 2014 at 5:22 PM, Vikram Garg <[email protected]>
> wrote:
>
>> Hello Xujun,
>> For the mixed FEM formulation, the pressure usually
>> lies in the L2 function space, whereas the velocity lies in the H1 function
>> space. These function spaces dictate the norm to be used for measuring the
>> error in the variable.
>>
>> The patch recovery estimator assumes an H1 error norm, and would need to
>> be told that it should use L2 for the pressure variable. Can you tell us
>> what the code is for building your patch recovery error estimator, I can
>> then tell you the necessary modifications.
>>
>> Thanks.
>>
>> On Mon, Oct 20, 2014 at 5:16 PM, Xujun Zhao <[email protected]> wrote:
>>
>>> The true solution, u has 1/r singularity and pressure should have 1/r^2
>>> singularity. So probably this is the case for singular problems, such as
>>> cracks.
>>>
>>> Yes, I limited the time of refinement in each element below 10 to make it
>>> stop. But the problem is that if the results do not converge, how can I
>>> trust my numerical results? is this the reason why 1/4 mid-point singular
>>> element is used in cracks?
>>>
>>> Xujun
>>>
>>> Xujun
>>>
>>> On Mon, Oct 20, 2014 at 5:07 PM, John Peterson <[email protected]>
>>> wrote:
>>>
>>> > On Mon, Oct 20, 2014 at 3:31 PM, Xujun Zhao <[email protected]> wrote:
>>> > > Hi all,
>>> > >
>>> > > For adaptive mesh refinement, an error estimator has to be used to
>>> > > approximately evaluate the errors in each element in order to
>>> determine
>>> > > which element is going to be refined. My question is which error
>>> > estimator
>>> > > should be used in the mixed FEM formulations, for example, Stokes
>>> > equation
>>> > > with u-p formulation. I read Bathe KJ's review paper:
>>> > > A posteriori error estimation techniques in practical finite element
>>> > > analysis, Computer & Structures 2005 p 235-265.
>>> > > in which they suggested another local error indicator.
>>> > >
>>> > > I did try kelly and patch recovery, and they are useful to locate the
>>> > > singular region and refine the mesh near the singularity. But the
>>> error
>>> > > norms become larger and larger with the refinement. It seems that the
>>> > > numerical solution is not convergent with mesh refinement.
>>> >
>>> > Is your true solution (especially for pressure) in H1?
>>> >
>>> > If not, like for the non-leaky lid-driven cavity, then Kelly will keep
>>> > giving you larger and larger error estimates.
>>> >
>>> > In practice, you can limit the amount of refinement in the singularity
>>> > by calling max_h_level() on the MeshRefinement object.
>>> >
>>> > --
>>> > John
>>> >
>>>
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>>
>>
>>
>> --
>> Vikram Garg
>> Postdoctoral Associate
>> Center for Computational Engineering
>> Massachusetts Institute of Technology
>> http://web.mit.edu/vikramvg/www/
>>
>> http://www.runforindia.org/runners/vikramg
>>
>
>
--
Vikram Garg
Postdoctoral Associate
Center for Computational Engineering
Massachusetts Institute of Technology
http://web.mit.edu/vikramvg/www/
http://www.runforindia.org/runners/vikramg
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