Thank you for your explanations. Basically I formed a weak form of the PDE for
one element and numerically integrate it at the Gaussian points based on the
interpolation from the local nodes. Subsequently, I assemble the weak forms
from all elements into a global system matrix based on a local-to-global
mapping of the nodes. After applying the boundary conditions, I solve this
linear system using a linear solver in Intel mkl.
In deal.II, I took almost the same procedures. I formed the weak form for each
element and assemble them based on the local-to-global mapping. After applying
the boundary conditions, I solved it with the SparseDirectFUMPK solver. The
only difference is that I got a very clear discontinuous gradient over the
edges of elements. I think the solution from deal.II is more close to the
theorem of FEM, which explains the fundamental idea of weak form. However, I am
puzzled at why I did not read it in the FEM books before.
Best
David
________________________________
From: Wolfgang Bangerth <bange...@colostate.edu>
Sent: Saturday, January 18, 2020, 11:58 PM
To: David Eaton; dealii@googlegroups.com
Subject: Re: [deal.II] discontinous contour over elements
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.
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www: http://www.math.colostate.edu/~bangerth/
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