On 1/17/20 9:11 PM, David Eaton wrote:
> 
> Thanks the help from you and the others. The issue of discontinuous vorticity 
> field is resolved. Theoretically, I understand the gradient should be 
> discontinuous for C0 elements.  However, I still want to convince myself with 
> a explanation. while I used Q2Q1 Taylor-Hood element, the discontinuous 
> contour still appear. What could be the reason for this?

And they should! You might be thinking that for Q2 functions, if you take the 
gradient, you should still get a linear function that is continuous. But the 
Q2 functions are only quadratic on each cell separately; they still have a 
kink at cell interfaces, and consequently their gradient is discontinuous.


> On the other hand, I 
> have a very simple FEM code using C0 elements without doing the projections.  
> I just simply assembly the matrix, use a Lagrangian shape function in C0 
> element and solve it with a linear solver. It does give a continuous contour 
> without doing a projection. What could be the theoretical reason why it does 
> not give a discontinuous contour? Is L2 projection is necessary step while 
> computing gradients over elements for C0 elements?

Can you describe in mathematical terms what you do? When you say "assemble a 
matrix" and "solve it", that sounds a lot like a projection to me.

Best
  W.

-- 
------------------------------------------------------------------------
Wolfgang Bangerth          email:                 bange...@colostate.edu
                            www: http://www.math.colostate.edu/~bangerth/

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