Sean:
It is hard to tell what exactly is going wrong, but before trying to bark up a
tree that is perhaps the wrong one, here are two questions:
* For the case of adaptive refinement, the solution looks discontinuous also
at edges without refinement. Is that reasonable? What's the element you are using?
* One of the things worth keeping in mind is that what you are visualizing is
a piecewise linear representation of the solution. However, I suspect you are
using a higher polynomial finite element. In other words, what you see in
Visit is not what the solution *actually* is. Take a look at the documentation
of DataOut::build_patches() for a discussion of how this can be addressed by
using subdivisions of the mesh to get a closer representation of what you are
computing.
Best
W.
On 8/17/24 07:26, Sean Carney wrote:
*** Caution: EXTERNAL Sender ***
Hi all,
I recently implemented a symmetric interior penalty DG method for a 4th order
problem with an interface.
I took as my "base code" step 74 (thank you for that!) and modified the
assemble, estimator, and energy_norm functions to suit my needs.
Unfortunately, I'm seeing some troubling results when doing local refinement.
A simple example of the unexpected behavior that I'm seeing can be seen in the
attached images.
Here the analytic solution to my problem posed on the (2d) unit cube is a
piecewise polynomial. It depends only on the horizontal variable $x$, and for
$x < 1/2$, the solution $u = 0$. For $x > 1/2$, the solution equals: [quintic
polynomial]*[exp(x)].
On a /uniform /coarse mesh with 4x4 grid cells, the computed solution is
already decent--in the eyeball norm, it is reasonable, and the error measured
in my "energy" norm (a kind of $H^2$ norm) is on the order of 1e-1.
However, if I refine just one cell in the mesh, the computed solution is
obviously incorrect to the eye, and the energy norm error is correspondingly
large (~2e1).
Whatever is causing this behavior is, I believe, also causing me to to see
incorrect rates of convergence upon a large number of local refinements. There
may be trouble in the "estimate" and "energy_norm" functions, but the attached
images suggest there is definitely trouble in the "assemble" function (I'm
using a direct solver, so nothing here is related to iterative linear solvers).
In contrast, if I refine globally, I get nice convergence rates--consistent
(or slightly better than) with what I expect from theory.
Additionally, I also am (of course) observing nice rates from the Step-74 code
upon local refinement. Both for the L-shaped domain with a corner and for the
smooth problem on the unit square.
I have been slowly loosing my mind trying to figure out what sort of error
I've introduced to make things not work, but can't find anything--for example,
all of the logic for working with jumps and averages across faces when hanging
nodes are present, /I think/, is handled by the MeshWorker framework.
Does anyone have any suggestions for things to try? I greatly appreciate any
input!
Thank you--
--Sean
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www: http://www.math.colostate.edu/~bangerth/
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