Getting Problem to get good convergence rate in Proximal Discontinuous Galerkin method of Elliptic obstacle problem

General Discussion
1

Respected Professors and experts,

I am trying to get a good convergence rate using the Proximal Discontinuous Galerkin(PDG) method for the elliptic obstacle problem.

Here is my main scheme:

PDG_scheme
PDG_scheme996×968 161 KB

Data for numerical experiments:

image
image1115×428 61.6 KB

image

Here, is my IPDG proximal galerkin code:

PDG_elliptic.edp (4.0 KB)

Note: The code runs correctly, but the results are not satisfactory. Additionally, how can the following space be defined in the code for the proximal EG method?.

image
image1072×130 21.3 KB

(How to write this Vh space in the FF++ code for Proximal EG method ?.)

Here is the full paper of proximal discontinuous Galerkin method of elliptic obstacle problem:

Proximal_dg_2026.pdf (503.2 KB)

I would be very grateful for any suggestions to obtain the expected order of convergence, i.e., 2, in the L^2-norm.

2

Probably you need to define a new fe space for (26) (see the FF++ doc to define new fe spaces, but it is rather difficult).
Independently of that, it seems that there is a bug regarding the example since the exact solution does not vanish on the boundary. I think that the domain should be the unit disc instead of the square, or otherwise with the square we have to set the BC u=u_ex.

3

Here is a version with correct Newton algorithm, for the unit disk and conforming FE (P1,P0).
pg_disk.edp (3.6 KB)

To compare with, here is a code for the semi-exact method
obstacle-semiexact.edp (4.4 KB)
The method is described in

(see appendix A for the general formulation, that is applicable to the obstacle problem with G=Id, and h the indicator function (in the sense of convex analysis) of the set of admissible unknowns u(x) satisfying u\geq\phi a.e.)

4

Sir, why is the order initially coming out to be very low?.

Results:

image
image976×240 7.28 KB

Thank you so much, Sir, for your constant guidance and kind support. I truly appreciate your time, patience, and valuable suggestions. Your help means a lot to me.

5

The theoretical order of accuracy is in the limit h\to0, thus we cannot say anything for ‘not so small’ h.

6

Okay Sir. The order is also increasing, as I observe, when N increases, where h=1/N.

7

Sir, I am wondering why applying the Symmetric Interior Penalty method to the continuous Galerkin code is giving optimized error estimates.

PDG_FF_Sir.edp (3.8 KB)

image
image1383×415 11.6 KB

8

It means the scheme blows up, I don’t know why. You have to check the formulation (fespace, bilinear form, etc)

9

Okay Sir. I will check and let you know.

10

If you put a minus sign on the second term:

			+ intalledges(Th)(//  loop on all  edge of all triangle 
                       ( jump(v)*mean(dn(u))-jump(u)*mean(dn(v)) 
                      + pena*jump(u)*jump(v)) / nTonEdge
                        )

it converges (I am not an expert on DG methods and I am not able to explain this).

11

Thank you so much, Sir. I have solved this issue. But I have seen that you considered alpha_k=1/k, whereas in my case alpha_k=2^(k-1). Although, the code is working fine when setting alpha_k=2^(k-1).

12

I don’t know how to determine the “best choice” for \alpha_k. The paper says “an unsummable sequence of positive proximity parameters \alpha_k”. Probably the choice influences the number of iterations to achieve convergence.

13

Yes Sir. Thanks for your guidance.