1. If I make delta larger(left: delta=0.05), or refine the mesh much
finer (right:"triangulation.refine_global (9);" while the original is
"triangulation.refine_global (7);") then I can get the results without
too much oscillations:
6.jpg
Yes, that is not surprising. You get oscillations if the mesh is not
fine enough to resolve the features of the solution. This is the same as
if you do a Fourier series of a function with a steep gradient: the sum
of the first few Fourier terms is oscillatory unless you include
sin(n*x) and cos(n*x) with n large enough for the sines and cosines to
have a period on the same order as the width of the gradient of the
original function.
You can therefore avoid the oscillations if either (i) you make the
steep gradient spread out over a larger length scale, (ii) increase n.
So, my questions are:
1.If I need to get
4.JPG
in every time step, which means that phi_x and phi_y are updated in
every time step, and the information of phi_x and phi_y are calculated
from "phi" by using function: "get_function_gradients".
And I don't know the exact value of phi and grad_phi in every time step.
So how can I remove oscillations?
I don't understand this. If you don't know what grad phi is, how do you
want to compute atan2(phi_y,phi_x) and from that F? If you don't know
what phi is, you also can't compute grad phi via a finite different
approximation.
It doesn't matter whether you compute the gradient exactly or via a FDM
approximation *conceptually*, but it will almost certainly be *more
accurate* to use the exact derivative with get_function_gradients().
If I can use FDM above in every time step, how to get real coordinates
of grid nodes, then get the real coordinates of adjacent nodes, and get
the values of phi_up, phi_down, phi_left, phi_right?
We generally do this by using FEValues::get_function_values() and
FEValues::quadrature_point(), which when initialized on a cell gives you
the values and locations of a function evaluated at quadrature points.
If you want the values at the center of a cell, you can always use
QMidpoint as your quadrature formula.
But, I will mention, that this approach only works if you have a uniform
mesh aligned with the x- and y-axes. That's not a very general case. I
would try to use a method that works for any geometry.
2. If this method can not go, how can I remove oscillations? Maybe use
DG to describe F? and add some interior penalty?
The problem is generic: If the function has a sharp gradient, then a
projection onto *any* finite dimensional space will yield oscillations
*unless* you can resolve the width of the gradient with your mesh. There
is simply nothing very much you do about this with linear operations.
In your case, you can at least prevent the propagation of oscillations
to neighboring cells if you project onto a DG space. You do not need
penalty terms since the equation you want to solve is local and has no
derivatives. Just do the projection locally.
Best
W.
--
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Wolfgang Bangerth email: bange...@colostate.edu
www: http://www.math.colostate.edu/~bangerth/
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