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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: Mon, 31 Dec 2018 10:11:59 +0100

Dear Yves Renard,
Thank you so much for your answers. The alternative to the Ghost penalty is a good idea to solve with P_k FEMs. For the remark about the slice, I don't use it to calculate the error, but just to export the solution in the physical domain. To calculate the error I use exactly the integration method with  MeshLevelSet as follow:
ls = gf.LevelSet(m,2,'(x-{X0})*(x-{X0}) + (y-{Y0})*(y-{Y0}) - {R}*{R}'.format(X0=x0, Y0=y0, R=r))
mls = gf.MeshLevelSet(m)
mim = gf.MeshIm('levelset',mls,'INSIDE', gf.Integ('IM_TRIANGLE(6)'))
#After creating the model and solve it, I calculate the error as follows:#
    L2error = gf.compute(mfu, Uap-Uex, 'L2 norm', mim)
    H1error = gf.compute(mfu, Uap-Uex, 'H1 norm', mim)
where mfu is given by:
mfu   = gf.MeshFem(m, 1)
But I still not able to get the optimal rate of convergence (I get approximately the same error for NX=16, 32, 64,etc. But if I take NX=400 for example, I get an approximate solution very similar to the exact solution with a very small error where the maximum value=2.1e-06). 
Thank you for the help.

Le jeu. 27 déc. 2018 à 09:54, Yves Renard <address@hidden> a écrit :

Dear Yassine,

Concerning the Ghost penalty, I am not sure that penalization of normal derivative is not sufficient even for P2 or P3 method. I think it should be sufficient.

Note also that an alternative to Ghost penalty is to directly interpolate the normal derivative in a neighbour element for elements with a small intersection with the domain as we proposed in the paper

J. Haslinger, Y. Renard. A new fictitious domain approach inspired by the extended finite element method. Siam J. on Numer. Anal., 47(2):1474--1499, 2009.

Best regards,


----- Original Message -----
From: "yves renard" <address@hidden>
To: "Yassine ZAIM" <address@hidden>
Cc: "getfem-users" <address@hidden>
Sent: Wednesday, December 26, 2018 10:02:11 PM
Subject: Re: [Getfem-users] Calculate the error in the physical domain

Dear Yassine,

Once again, the slice operations are only for graphical post-processing. You cannot compute an error with them. You have to use cut integration methods instead.

Best regards,


----- Original Message -----
From: "Yassine ZAIM" <address@hidden>
To: "yves renard" <address@hidden>
Cc: "getfem-users" <address@hidden>
Sent: Wednesday, December 19, 2018 5:00:01 PM
Subject: Re: [Getfem-users] Calculate the error in the physical domain

Dear Yves Renard and getfem++ users,
Thank you so much for your answer. I am inspired by the
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
> ""
> 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)
>   INSA-Lyon
>   20, rue Albert Einstein
>   69621 Villeurbanne Cedex, FRANCE
> ---------

*ZAIM Yassine *
*PhD in Applied Mathematics*

ZAIM Yassine 
PhD in Applied Mathematics

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