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## Re: [Getfem-users] Calculate the error in the physical domain

 From: Yassine ZAIM Subject: Re: [Getfem-users] Calculate the error in the physical domain Date: Wed, 19 Dec 2018 17:00:01 +0100

Dear Yves Renard and getfem++ users,
Thank you so much for your answer. I am inspired by the "demo_fictitious_domains.py" example to achieve my test and the proposed method is what I did at the beginning. But I get the same value of the error for different values of NX and I can't get the optimal order. For this, I am saying maybe the outside domain of Omega which creates the problem, for this I asked how to restrict the computation of error to Omega. I enclosed my code if you can see it quickly, maybe you will find out the mistake.
I have another question about the Ghost penalty. For the P_2 element, I guess that I have to work with the hessian "hess_u" and "hess_Test_u" But how we can implement it for P_3, P_4,... spaces.
Thank you so much for your help.

Le mer. 19 déc. 2018 à 14:39, Yves Renard <address@hidden> a écrit :

Dear Yassine,

The slice operation is only for graphical post-processing. You cannot use a sliced solution into computations because the obtained vector represent the value of the solution on a cut mesh representing the computed intersection. The obtained vector is no longer a dof vector on the meshfem mfu. The standard way to compute this error is to produce a cut integration method, and just compute the L2 or H1 norms in a standard way with it. You can produce cut integration method with the MeshLevelSet and MeshIM('levelset') objects. An example of use in the python test program "demo_fictitious_domains.py"

Best regards,

Yves

Le 18/12/2018 à 18:54, Yassine ZAIM a écrit :
Dear getfem++ users,
I am trying to learn how to implement the fictitious domain method for the simple problem of Poisson. My domain of interest Omega is a circle of center (x0=0,y0=0) and radius r=0.35. I made the resolution in the fictitious domain (the square [-0.5, 0.5]^2) and after that, I used the slice to restrict my solution to the physical domain.
sl = gf.Slice(('comp',('ball', +1, [x0, y0], r)), m, 5)
sl.export_to_vtk('App_Solution.vtk', 'ascii', mfu, Uap)
sl.export_to_vtk('Exc_Solution.vtk', 'ascii', mfu, Uex)
sl.export_to_vtk('Error-Ex-App.vtk', 'ascii', mfu, Uap-Uex)
When I see the approximate and the exact solution in addition to the error I can say that I get a good approximate solution. But I am trying to get the optimal order of convergence (h and h^2) like in the papers "J. Haslinger and Y. Renard" or "E. Burman and P. Hansbo". So I interpolate the exact and approximate solution in the slice (domain of interest).
Ue = gf.compute(mfu, Uex, 'interpolate on', sl)
U = gf.compute(mfu, Uap, 'interpolate on', sl)
And after that I tried to calculate the error for different values of NX=[16,32,..]
L2error = gf.compute(mfu, U-Ue, 'L2 norm', mim)
H1error = gf.compute(mfu, U-Ue, 'H1 norm', mim)
such that :
mim = gf.MeshIm('levelset',mls,'inside', gf.Integ('IM_TRIANGLE(6)'))
and  mfu.set_fem(gf.Fem('FEM_PK(2,1)'))
But I get the following error in the "compute" function (of the errors) :
return getfem('compute', mf, U, what, *args)
RuntimeError: (Getfem::InterfaceError) -- The trailing dimension of argument 2 (an array of size 4670) has 4670 elements, 289 were expected.
My question is how to calculate the error just in the interesting domain (physical domain, slice sl in my case).
Thank you in advance.

--
ZAIM Yassine
PhD in Applied Mathematics

```--

Yves Renard (address@hidden)       tel : (33) 04.72.43.87.08
INSA-Lyon
20, rue Albert Einstein
69621 Villeurbanne Cedex, FRANCE
http://math.univ-lyon1.fr/~renard

---------
```

--
ZAIM Yassine
PhD in Applied Mathematics

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