[dune-pdelab] Computing Maximum and Average solution with DG

[Jared Okiro]

Thanks Jö for the input.
-- 
Jared Ouma Okiro
Institute for Analysis and Numerics
Otto von Guericke Universität
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Quoting Jö Fahlke <jorrit at jorrit.de>:

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> Am Sat, 19. May 2012, 20:10:00 +0200 schrieb Steffen Müthing:
>> I think the problem is due to the different interpolation schemes  
>> used by the
>> two methods (your function and the VTKWriter, respectively):
>>
>> You use a first order DG scheme and calculate the integration order in your
>> function as
>>
>> max(2*(dg_order -1),0),
>>
>> which yields 0 in your case, so you get a zero order integration  
>> rule which only
>> evaluates your solution at the cell center. For the VTKWriter, you  
>> use a vertex
>> data set (and the subsampling level is 0), so the VTKWriter evaluates your
>> solution at the vertices. And your solution assumes its maximum  
>> only at vertex
>> positions...
>>
>> Try increasing the integration order in your function and see if that helps.
>
> That should improve things but won't yield the exact maximum.  Gaussian
> quadrature points are never at the element boundaries, but the maximum for
> linear shape functions will always be in an element's corner.
>
> For first order shape functions, to get the exact maximum, just evaluate the
> solution at the elements' corners.  The generic reference elements can tell
> you the local coordinates of the corners.

In the test case that I attached, I presume that I am evaluating the  
maximum solution at the element's corners. If this is not correct then  
please help me on how I can achieve this. I also presume that the DG  
shape functions for P1 elements live at the element's corners(I have  
not verified this exactly). You have mentioned that one can determine  
the local coordinates of the corners for the generic reference  
elements, please elaborate this coz I seem not to understand it well.
>
> Do you need the maximum for shape functions of higher order?  If so, can you
> live with the approximation you get from gaussian quadrature points?  I don't
> know of an easy the to get the exact maximum in this case...

Yes I need to compute the maximum solution for my system with higher  
order shape functions(upto order 5). Is there a unique way of  
computing the maximum solution for higher order shape functions?. I  
will appreciate more advice on this.
Thank you.
Jared.
>
> Bye,
> Jö.
>
> --
> Jorrit (Jö) Fahlke, Interdisciplinary Center for Scientific Computing,
> Heidelberg University, Im Neuenheimer Feld 368, D-69120 Heidelberg
> Tel: +49 6221 54 8890 Fax: +49 6221 54 8884
>
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