# Getting Problem to get good convergence rate in Proximal Discontinuous Galerkin method of Elliptic obstacle problem
**URL:** https://community.freefem.org/t/getting-problem-to-get-good-convergence-rate-in-proximal-discontinuous-galerkin-method-of-elliptic-obstacle-problem/4264
**Category:** General Discussion
**Created:** [April 30, 2026, 11:24am UTC](https://community.freefem.org/t/getting-problem-to-get-good-convergence-rate-in-proximal-discontinuous-galerkin-method-of-elliptic-obstacle-problem/4264 "2026-04-30T11:24:21Z")
**Posts on this page:** 13
**Page:** 1
### Author:  [@Monirul25](https://community.freefem.org/u/Monirul25)
#### Post date: [April 30, 2026, 11:24am UTC](https://community.freefem.org/t/getting-problem-to-get-good-convergence-rate-in-proximal-discontinuous-galerkin-method-of-elliptic-obstacle-problem/4264/1 "2026-04-30T11:24:21Z")
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:
**Data for numerical experiments:**
**Here, is my IPDG proximal galerkin code:**
[PDG\_elliptic.edp](https://community.freefem.org/uploads/short-url/sz2dT0AiwkBTf59FVUq8h7IqUsU.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?.**
( **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](https://community.freefem.org/uploads/short-url/mBMUlKDOtZaPQOqRjSs4GCECXzf.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.
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### Author:  [@fb77](https://community.freefem.org/u/fb77)
#### Post date: [May 6, 2026, 3:32pm UTC](https://community.freefem.org/t/getting-problem-to-get-good-convergence-rate-in-proximal-discontinuous-galerkin-method-of-elliptic-obstacle-problem/4264/2 "2026-05-06T15:32:33Z")
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.
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### Author:  [@fb77](https://community.freefem.org/u/fb77)
#### Post date: [May 11, 2026, 2:05pm UTC](https://community.freefem.org/t/getting-problem-to-get-good-convergence-rate-in-proximal-discontinuous-galerkin-method-of-elliptic-obstacle-problem/4264/3 "2026-05-11T14:05:52Z")
Here is a version with correct Newton algorithm, for the unit disk and conforming FE (P1,P0).
[pg\_disk.edp](https://community.freefem.org/uploads/short-url/qB4NYyp84WEWZYy1LpmSQcTErHC.edp) (3.6 KB)
To compare with, here is a code for the semi-exact method
[obstacle-semiexact.edp](https://community.freefem.org/uploads/short-url/6m5fvVhD5aJvzRXh02hrjEKrJp1.edp) (4.4 KB)
The method is described in
> **[A semi-exact primal-dual method for steady viscoplastic flows](https://smai-jcm.centre-mersenne.org/articles/10.5802/jcm.145/)**
(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.)
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### Author:  [@Monirul25](https://community.freefem.org/u/Monirul25)
#### Post date: [May 11, 2026, 3:27pm UTC](https://community.freefem.org/t/getting-problem-to-get-good-convergence-rate-in-proximal-discontinuous-galerkin-method-of-elliptic-obstacle-problem/4264/4 "2026-05-11T15:27:12Z")
Sir, **why is the order initially coming out to be very low**?.
Results:
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.
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### Author:  [@fb77](https://community.freefem.org/u/fb77)
#### Post date: [May 11, 2026, 3:29pm UTC](https://community.freefem.org/t/getting-problem-to-get-good-convergence-rate-in-proximal-discontinuous-galerkin-method-of-elliptic-obstacle-problem/4264/5 "2026-05-11T15:29:57Z")
The theoretical order of accuracy is in the limit h\to0, thus we cannot say anything for ‘not so small’ h.
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### Author:  [@Monirul25](https://community.freefem.org/u/Monirul25)
#### Post date: [May 11, 2026, 3:37pm UTC](https://community.freefem.org/t/getting-problem-to-get-good-convergence-rate-in-proximal-discontinuous-galerkin-method-of-elliptic-obstacle-problem/4264/6 "2026-05-11T15:37:28Z")
Okay Sir. The order is also increasing, as I observe, when N increases, where h=1/N.
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### Author:  [@Monirul25](https://community.freefem.org/u/Monirul25)
#### Post date: [May 11, 2026, 4:31pm UTC](https://community.freefem.org/t/getting-problem-to-get-good-convergence-rate-in-proximal-discontinuous-galerkin-method-of-elliptic-obstacle-problem/4264/7 "2026-05-11T16:31:01Z")
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](https://community.freefem.org/uploads/short-url/gm9O4SHZiQx0clbwm6xmtiKeAJ5.edp) (3.8 KB)
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### Author:  [@fb77](https://community.freefem.org/u/fb77)
#### Post date: [May 11, 2026, 5:12pm UTC](https://community.freefem.org/t/getting-problem-to-get-good-convergence-rate-in-proximal-discontinuous-galerkin-method-of-elliptic-obstacle-problem/4264/8 "2026-05-11T17:12:59Z")
It means the scheme blows up, I don’t know why. You have to check the formulation (fespace, bilinear form, etc)
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### Author:  [@Monirul25](https://community.freefem.org/u/Monirul25)
#### Post date: [May 11, 2026, 6:11pm UTC](https://community.freefem.org/t/getting-problem-to-get-good-convergence-rate-in-proximal-discontinuous-galerkin-method-of-elliptic-obstacle-problem/4264/9 "2026-05-11T18:11:16Z")
Okay Sir. I will check and let you know.
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### Author:  [@fb77](https://community.freefem.org/u/fb77)
#### Post date: [May 12, 2026, 8:46am UTC](https://community.freefem.org/t/getting-problem-to-get-good-convergence-rate-in-proximal-discontinuous-galerkin-method-of-elliptic-obstacle-problem/4264/10 "2026-05-12T08:46:27Z")
If you put a minus sign on the second term:
```auto
+ 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).
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### Author:  [@Monirul25](https://community.freefem.org/u/Monirul25)
#### Post date: [May 12, 2026, 9:17am UTC](https://community.freefem.org/t/getting-problem-to-get-good-convergence-rate-in-proximal-discontinuous-galerkin-method-of-elliptic-obstacle-problem/4264/11 "2026-05-12T09:17:02Z")
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).
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### Author:  [@fb77](https://community.freefem.org/u/fb77)
#### Post date: [May 12, 2026, 10:17am UTC](https://community.freefem.org/t/getting-problem-to-get-good-convergence-rate-in-proximal-discontinuous-galerkin-method-of-elliptic-obstacle-problem/4264/12 "2026-05-12T10:17:59Z")
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.
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### Author:  [@Monirul25](https://community.freefem.org/u/Monirul25)
#### Post date: [May 12, 2026, 1:30pm UTC](https://community.freefem.org/t/getting-problem-to-get-good-convergence-rate-in-proximal-discontinuous-galerkin-method-of-elliptic-obstacle-problem/4264/13 "2026-05-12T13:30:51Z")
Yes Sir. Thanks for your guidance.