# Getting Problem to get Superconvergence results in Hybrid Higher Order (HHO) Method in Poisson Equation **URL:** \ **Category:** General Discussion\ **Created:** [March 18, 2026, 10:06am UTC](https://community.freefem.org/t/getting-problem-to-get-superconvergence-results-in-hybrid-higher-order-hho-method-in-poisson-equation/4242 "2026-03-18T10:06:52Z")\ **Posts on this page:** 16\ **Page:** 1
**Author:** ![Monirul25](https://yyz2.discourse-cdn.com/flex030/user_avatar/community.freefem.org/monirul25/32/3287_2.png) [@Monirul25](https://community.freefem.org/u/Monirul25)\ **Post date:** [March 18, 2026, 10:06am UTC](https://community.freefem.org/t/getting-problem-to-get-superconvergence-results-in-hybrid-higher-order-hho-method-in-poisson-equation/4242/1 "2026-03-18T10:06:52Z")
Hello, all experts and respected professors, I am facing a problem getting **superconvergence results** ( **Need to get (k+2) order in L^2-norm and (k+1) in reconstruction gradient using P^k polynomial approx.** ) for the Poisson equation by using **HHO** in FF++. Here is the HHO scheme using the reconstruction operator : ![eq1](https://canada1.discourse-cdn.com/flex030/uploads/freefem/original/2X/e/e31afa8e53ecb4d97e346406c50f9ea35b0e9737.png) ![eq2](https://canada1.discourse-cdn.com/flex030/uploads/freefem/original/2X/d/d6d2054afca7de6b1b31ad01c6f1ce4d70f16c5c.png) **Note that gamma=1 (in equation (2)).** **Details paper:** [Very\_good.pdf](https://community.freefem.org/uploads/short-url/4dEFJkt3FpCHB2U3LH0fsSEv2sF.pdf) (250.3 KB) **My Code:** [HHO\_best.edp](https://community.freefem.org/uploads/short-url/eVhXfmLwQ94mlrTuDmATAwM8zY6.edp) (3.2 KB) **Results:** ![results](https://canada1.discourse-cdn.com/flex030/uploads/freefem/original/2X/a/a04f78517ae2f32cc0292810427c9bd0dba7f9a4.png) **But i need 3 order in L2-norm and 2 order in reconstruction gradient. Why is the Superconvergence not coming??.** I really appreciate any help I get. Thanks in advance !. ---
**Author:** ![fb77](https://yyz2.discourse-cdn.com/flex030/user_avatar/community.freefem.org/fb77/32/3796_2.png) [@fb77](https://community.freefem.org/u/fb77)\ **Post date:** [March 21, 2026, 3:44pm UTC](https://community.freefem.org/t/getting-problem-to-get-superconvergence-results-in-hybrid-higher-order-hho-method-in-poisson-equation/4242/2 "2026-03-21T15:44:41Z")
Dear Monirul, In this problem you have to repeatedly use Lagrange multipliers to solve all the equations simultaneously. Code: [HHO.edp](https://community.freefem.org/uploads/short-url/u1s27zxtJJFSprUzB6deV5CrEMr.edp) (4.9 KB) Francois. ---
**Author:** ![Monirul25](https://yyz2.discourse-cdn.com/flex030/user_avatar/community.freefem.org/monirul25/32/3287_2.png) [@Monirul25](https://community.freefem.org/u/Monirul25)\ **Post date:** [March 22, 2026, 4:03pm UTC](https://community.freefem.org/t/getting-problem-to-get-superconvergence-results-in-hybrid-higher-order-hho-method-in-poisson-equation/4242/3 "2026-03-22T16:03:25Z")
Thank you so much Sir for your support. **But as per the paper, the L^2-norm order is fine, but order 2 should come for reconstruction gradient if I am not wrong.** Pls see: ![image](https://canada1.discourse-cdn.com/flex030/uploads/freefem/original/2X/d/d17fa9aa875ee218a93e54845be1abe1ae5f9cca.png) ![image](https://canada1.discourse-cdn.com/flex030/uploads/freefem/original/2X/1/1d0fa6c2184cccf51e244b40d186cabe82520830.png) ![image](https://canada1.discourse-cdn.com/flex030/uploads/freefem/original/2X/0/00aa807fd38b86edd8baad61817de28deb944332.png)(Results that are coming from your corrected version code. My question is that RGerrors corresponding order should be 2, sir??. ) ---
**Author:** ![fb77](https://yyz2.discourse-cdn.com/flex030/user_avatar/community.freefem.org/fb77/32/3796_2.png) [@fb77](https://community.freefem.org/u/fb77)\ **Post date:** [March 22, 2026, 5:29pm UTC](https://community.freefem.org/t/getting-problem-to-get-superconvergence-results-in-hybrid-higher-order-hho-method-in-poisson-equation/4242/4 "2026-03-22T17:29:37Z")
You are right, the grad error was not computed correctly. Correction: [HHO.edp](https://community.freefem.org/uploads/short-url/luJF53M56lke4wllT2N63hIGGpX.edp) (5.1 KB) ---
**Author:** ![Monirul25](https://yyz2.discourse-cdn.com/flex030/user_avatar/community.freefem.org/monirul25/32/3287_2.png) [@Monirul25](https://community.freefem.org/u/Monirul25)\ **Post date:** [March 22, 2026, 5:43pm UTC](https://community.freefem.org/t/getting-problem-to-get-superconvergence-results-in-hybrid-higher-order-hho-method-in-poisson-equation/4242/5 "2026-03-22T17:43:28Z")
Many many thanks Sir. ---
**Author:** ![fb77](https://yyz2.discourse-cdn.com/flex030/user_avatar/community.freefem.org/fb77/32/3796_2.png) [@fb77](https://community.freefem.org/u/fb77)\ **Post date:** [March 27, 2026, 2:49pm UTC](https://community.freefem.org/t/getting-problem-to-get-superconvergence-results-in-hybrid-higher-order-hho-method-in-poisson-equation/4242/6 "2026-03-27T14:49:10Z")
The method of Lagrange multipliers gives the solution, but is indeed not the way that should be chosen for hybrid methods, because it leads to a very large linear system. Instead one should use reduction, by inversion of the local operators. In order to show how to implement an hybrid method with inversion of the local problems in FreeFem++, let me explain how to do on this example problem of the Laplace operator by the hybrid high order method of DiPietro and Ern. **1. Discontinuous spaces** Let me recall that Pkdc (with k an integer) is the space of functions that are polynomial of degree at most k in each triangle, without any continuity requirement from one triangle to another. Similarly, Pkedge is the space of functions that are defined only on edges of the mesh, and that are polynomial of degree at most k in each edge, without any continuity requirement from one edge to another. Pkedgedc is similar to Pkedge except that we have two polynomial functions per edge, one for each side of the edge (or only 1 if the edge is a boundary edge), hence two different values when approaching the edge from one neighbouring triangle or the other. **2. The scheme** ![hybrid-DPE](https://canada1.discourse-cdn.com/flex030/uploads/freefem/original/2X/7/7be60fcef9d544906000a07a9085f2dc16d25e4f.png) **3. The reduced formulation** ![hybrid-DPEsolve](https://canada1.discourse-cdn.com/flex030/uploads/freefem/original/2X/5/54a4abc0a92009eb9561c3d44a9fec04f5534e45.png) **4. FreeFem++ code** [HHO-reduced.edp](https://community.freefem.org/uploads/short-url/rkzhSUSEZUZEshkTZ7fqiH5B8IL.edp) (14.5 KB) ---
**Author:** ![Monirul25](https://yyz2.discourse-cdn.com/flex030/user_avatar/community.freefem.org/monirul25/32/3287_2.png) [@Monirul25](https://community.freefem.org/u/Monirul25)\ **Post date:** [March 27, 2026, 6:05pm UTC](https://community.freefem.org/t/getting-problem-to-get-superconvergence-results-in-hybrid-higher-order-hho-method-in-poisson-equation/4242/7 "2026-03-27T18:05:41Z")
Dear Prof. François, I am very grateful to you for sharing this valuable information and code. It will greatly help me in my work. I truly appreciate the support you have provided earlier and now. Thank you very much. **Could you kindly share the details of the paper from which the screenshot images were taken?.** Best regards, Monirul ---
**Author:** ![fb77](https://yyz2.discourse-cdn.com/flex030/user_avatar/community.freefem.org/fb77/32/3796_2.png) [@fb77](https://community.freefem.org/u/fb77)\ **Post date:** [March 27, 2026, 6:06pm UTC](https://community.freefem.org/t/getting-problem-to-get-superconvergence-results-in-hybrid-higher-order-hho-method-in-poisson-equation/4242/8 "2026-03-27T18:06:54Z")
It is not a paper, this is just what I have written right now in order to explain the content of the code. ---
**Author:** ![Monirul25](https://yyz2.discourse-cdn.com/flex030/user_avatar/community.freefem.org/monirul25/32/3287_2.png) [@Monirul25](https://community.freefem.org/u/Monirul25)\ **Post date:** [March 27, 2026, 6:10pm UTC](https://community.freefem.org/t/getting-problem-to-get-superconvergence-results-in-hybrid-higher-order-hho-method-in-poisson-equation/4242/9 "2026-03-27T18:10:01Z")
Okay, thank you for your effort, Sir. ![image](https://canada1.discourse-cdn.com/flex030/uploads/freefem/original/2X/c/c2a8a4023c9809bbe173fb3b70743c8beed40d7e.png) (Results of Lagrange’s multiplier seemed better than HHO-reduced.edp Sir). **If I am not wrong, I hope both codes will work for time-dependent as well as higher-order equations, sir!.** ---
**Author:** ![fb77](https://yyz2.discourse-cdn.com/flex030/user_avatar/community.freefem.org/fb77/32/3796_2.png) [@fb77](https://community.freefem.org/u/fb77)\ **Post date:** [March 27, 2026, 6:12pm UTC](https://community.freefem.org/t/getting-problem-to-get-superconvergence-results-in-hybrid-higher-order-hho-method-in-poisson-equation/4242/10 "2026-03-27T18:12:42Z")
In the Lagrange multiplier code the mesh was `square()`, whereas in the last code it is `buildmesh()`. If you use the same mesh for both codes you should get the same results. ---
**Author:** ![Monirul25](https://yyz2.discourse-cdn.com/flex030/user_avatar/community.freefem.org/monirul25/32/3287_2.png) [@Monirul25](https://community.freefem.org/u/Monirul25)\ **Post date:** [March 27, 2026, 6:14pm UTC](https://community.freefem.org/t/getting-problem-to-get-superconvergence-results-in-hybrid-higher-order-hho-method-in-poisson-equation/4242/11 "2026-03-27T18:14:32Z")
Yes Sir. Thank you So much. **If I am not mistaken, I believe the ideas in both codes will work for time-dependent as well as higher-order equations, sir.** ---
**Author:** ![fb77](https://yyz2.discourse-cdn.com/flex030/user_avatar/community.freefem.org/fb77/32/3796_2.png) [@fb77](https://community.freefem.org/u/fb77)\ **Post date:** [March 27, 2026, 6:17pm UTC](https://community.freefem.org/t/getting-problem-to-get-superconvergence-results-in-hybrid-higher-order-hho-method-in-poisson-equation/4242/12 "2026-03-27T18:17:59Z")
For time dependent yes, but of course the computational time could be high. About high order, here it works for k=2 hence 4th order, it is already high. To make it work with higher k is difficult (quadrature formulas, issues with respect to the dofs in Pkdc that are slightly inside the triangle…). ---
**Author:** ![Monirul25](https://yyz2.discourse-cdn.com/flex030/user_avatar/community.freefem.org/monirul25/32/3287_2.png) [@Monirul25](https://community.freefem.org/u/Monirul25)\ **Post date:** [March 27, 2026, 6:19pm UTC](https://community.freefem.org/t/getting-problem-to-get-superconvergence-results-in-hybrid-higher-order-hho-method-in-poisson-equation/4242/13 "2026-03-27T18:19:21Z")
Okay thank you sir!. ---
**Author:** ![Monirul25](https://yyz2.discourse-cdn.com/flex030/user_avatar/community.freefem.org/monirul25/32/3287_2.png) [@Monirul25](https://community.freefem.org/u/Monirul25)\ **Post date:** [March 27, 2026, 6:57pm UTC](https://community.freefem.org/t/getting-problem-to-get-superconvergence-results-in-hybrid-higher-order-hho-method-in-poisson-equation/4242/14 "2026-03-27T18:57:03Z")
Dear Sir, Could you please share the appropriate reference for the **Lagrange multiplier used in HHO.edp**?. It would be very helpful if you could also provide a detailed explanation of it. Thank you very much for your support. ---
**Author:** ![fb77](https://yyz2.discourse-cdn.com/flex030/user_avatar/community.freefem.org/fb77/32/3796_2.png) [@fb77](https://community.freefem.org/u/fb77)\ **Post date:** [March 30, 2026, 3:01pm UTC](https://community.freefem.org/t/getting-problem-to-get-superconvergence-results-in-hybrid-higher-order-hho-method-in-poisson-equation/4242/15 "2026-03-30T15:01:40Z")
Here is the basics of the method of Lagrange multipliers. ![Lagrange-multipliers](https://canada1.discourse-cdn.com/flex030/uploads/freefem/original/2X/4/482465f0d8f2735511350775ffb8e199889e4275.png) The application to the problem (4)-(5) above is done as follows. ![HHO-Lag1](https://canada1.discourse-cdn.com/flex030/uploads/freefem/original/2X/f/f92fb07543801e96201db46a8c5ed9a8e86f4e34.png) ![HHO-Lag2](https://canada1.discourse-cdn.com/flex030/uploads/freefem/original/2X/5/5f12fea9ef5e8f6a98393ba8f2caa7cf4c3a7a07.png) ---
**Author:** ![Monirul25](https://yyz2.discourse-cdn.com/flex030/user_avatar/community.freefem.org/monirul25/32/3287_2.png) [@Monirul25](https://community.freefem.org/u/Monirul25)\ **Post date:** [March 30, 2026, 3:14pm UTC](https://community.freefem.org/t/getting-problem-to-get-superconvergence-results-in-hybrid-higher-order-hho-method-in-poisson-equation/4242/16 "2026-03-30T15:14:23Z")
Thank you so much Sir.