# Lagrange multiplier for Dirichlet boundary condition **URL:** \ **Category:** General Discussion\ **Created:** [April 18, 2023, 7:06am UTC](https://community.freefem.org/t/lagrange-multiplier-for-dirichlet-boundary-condition/2402 "2023-04-18T07:06:50Z")\ **Posts on this page:** 11\ **Page:** 2
**Author:** ![prj](https://avatars.discourse-cdn.com/v4/letter/p/ecae2f/32.png) [@prj](https://community.freefem.org/u/prj)\ **Post date:** [June 20, 2023, 2:40pm UTC](https://community.freefem.org/t/lagrange-multiplier-for-dirichlet-boundary-condition/2402/21 "2023-06-20T14:40:42Z")
Yes, since `Th` is distributed in the `buildDmesh()` call. ---
**Author:** ![yongxing](https://yyz2.discourse-cdn.com/flex030/user_avatar/community.freefem.org/yongxing/32/3632_2.png) [@yongxing](https://community.freefem.org/u/yongxing)\ **Post date:** [June 21, 2023, 6:28am UTC](https://community.freefem.org/t/lagrange-multiplier-for-dirichlet-boundary-condition/2402/22 "2023-06-21T06:28:43Z")
Hi, ```auto real[int] Af = af(0,Sh); Mat M=[[UU, UP', UA'], [UP, 0, 0], [UA, 0, 0]]; int undof = Vh.ndof; int pndof = Ph.ndof; int andof = Sh.ndof; int tndof = undof + pndof + andof; /* int undof = UU.n; int pndof = PP.n; int andof = AA.n; int tndof = undof + pndof + andof; */ real[int] f(tndof); f(0:undof + pndof - 1) = 0; f(undof + pndof : tndof - 1) = Af; real[int] fnew; ChangeNumbering(M, f, fnew); real[int] sol = M^-1*fnew; ``` Can I create my FreeFem vector `f` first, then use my **big martrix M** to convert it to a PETSc vector `fnew,` as done above, finally, solve it by` sol = M^-1*fnew;` ? ---
**Author:** ![prj](https://avatars.discourse-cdn.com/v4/letter/p/ecae2f/32.png) [@prj](https://community.freefem.org/u/prj)\ **Post date:** [June 21, 2023, 9:38am UTC](https://community.freefem.org/t/lagrange-multiplier-for-dirichlet-boundary-condition/2402/23 "2023-06-21T09:38:28Z")
You cannot with `M` because it’s missing some diagonal terms. However, see, e.g., `oseen-2d-PETSc.edp`, you could do `ChangeNumbering([UU, PP, AA], [rhsU, rhsP, rhsA], rhsPETSc);`. Also note that if you use the `^-1` syntax, your vector should be in FreeFEM numbering, not PETSc numbering, so there is no need for `ChangeNumbering()`… ---
**Author:** ![yongxing](https://yyz2.discourse-cdn.com/flex030/user_avatar/community.freefem.org/yongxing/32/3632_2.png) [@yongxing](https://community.freefem.org/u/yongxing)\ **Post date:** [June 21, 2023, 2:59pm UTC](https://community.freefem.org/t/lagrange-multiplier-for-dirichlet-boundary-condition/2402/24 "2023-06-21T14:59:18Z")
My matrix is a nested block matrix like ```auto Mat M=[[UU, UP', UA'], [UP, 0, 0], [UA, 0, 0]]; ``` Can I still use syntax `M^-1 *f` with f being FF numbering? I notice FF numbering has the overlapped nodes, which then has more degrees of freedom than PETSc numbering, is this OK? ---
**Author:** ![prj](https://avatars.discourse-cdn.com/v4/letter/p/ecae2f/32.png) [@prj](https://community.freefem.org/u/prj)\ **Post date:** [June 21, 2023, 3:09pm UTC](https://community.freefem.org/t/lagrange-multiplier-for-dirichlet-boundary-condition/2402/25 "2023-06-21T15:09:48Z")
Yes, it should be OK. Or you can use `KSPSolve(M, rhsPETSc, solPETSc)`. ---
**Author:** ![yongxing](https://yyz2.discourse-cdn.com/flex030/user_avatar/community.freefem.org/yongxing/32/3632_2.png) [@yongxing](https://community.freefem.org/u/yongxing)\ **Post date:** [June 21, 2023, 3:23pm UTC](https://community.freefem.org/t/lagrange-multiplier-for-dirichlet-boundary-condition/2402/26 "2023-06-21T15:23:11Z")
Thank you. I have tried this using two cores, but it has an error of ```auto malloc(): invalid size (unsorted) malloc(): invalid size (unsorted) ``` [stokes\_lagrange\_parallel\_4.edp](https://community.freefem.org/uploads/short-url/mstNwDv0nNVF0PalTGtX12WtiKA.edp) (3.8 KB) There is no problem of using one core. ---
**Author:** ![prj](https://avatars.discourse-cdn.com/v4/letter/p/ecae2f/32.png) [@prj](https://community.freefem.org/u/prj)\ **Post date:** [June 21, 2023, 3:34pm UTC](https://community.freefem.org/t/lagrange-multiplier-for-dirichlet-boundary-condition/2402/27 "2023-06-21T15:34:32Z")
What do you get with a FreeFEM compiled with debugging, i.e., what I told you to do since the beginning? ---
**Author:** ![prj](https://avatars.discourse-cdn.com/v4/letter/p/ecae2f/32.png) [@prj](https://community.freefem.org/u/prj)\ **Post date:** [June 21, 2023, 3:37pm UTC](https://community.freefem.org/t/lagrange-multiplier-for-dirichlet-boundary-condition/2402/28 "2023-06-21T15:37:50Z")
Just look at `stokes-block-2d-PETSc.edp`, it’s clear from the name passed to the argument `fields` that it should be a vector in PETSc numbering. Yet, your are supplying a vector in FreeFEM numbering… ---
**Author:** ![yongxing](https://yyz2.discourse-cdn.com/flex030/user_avatar/community.freefem.org/yongxing/32/3632_2.png) [@yongxing](https://community.freefem.org/u/yongxing)\ **Post date:** [June 21, 2023, 3:42pm UTC](https://community.freefem.org/t/lagrange-multiplier-for-dirichlet-boundary-condition/2402/29 "2023-06-21T15:42:44Z")
Yes, I tried to debug, but my machine does not allow me to do this as shown bellow ![MicrosoftTeams-image](https://canada1.discourse-cdn.com/flex030/uploads/freefem/original/2X/7/785723ecc00c0c2a8c58735ce87d3fe3f6b7868f.png) , sorry. ---
**Author:** ![yongxing](https://yyz2.discourse-cdn.com/flex030/user_avatar/community.freefem.org/yongxing/32/3632_2.png) [@yongxing](https://community.freefem.org/u/yongxing)\ **Post date:** [June 21, 2023, 4:04pm UTC](https://community.freefem.org/t/lagrange-multiplier-for-dirichlet-boundary-condition/2402/30 "2023-06-21T16:04:47Z")
Finally it works – FF vector for the right-hand side and PETSc vector for the fields. Thank you for all your help and your patience. I feel I have asked too many questions and I am sorry for this. I post the final version of this code here: [stokes\_lagrange\_parallel\_4.edp](https://community.freefem.org/uploads/short-url/oGkTzjFSyr28gIymHL5M6wXEn6E.edp) (3.9 KB) in case it will be useful for someone else. Thanks again for all your help @prj ! ---
**Author:** ![prj](https://avatars.discourse-cdn.com/v4/letter/p/ecae2f/32.png) [@prj](https://community.freefem.org/u/prj)\ **Post date:** [June 21, 2023, 4:06pm UTC](https://community.freefem.org/t/lagrange-multiplier-for-dirichlet-boundary-condition/2402/31 "2023-06-21T16:06:12Z")
No problem, glad it’s working. Hopefully the next questions will be more challenging! 🙂 [Previous page](https://community.freefem.org/t/lagrange-multiplier-for-dirichlet-boundary-condition/2402.md?page=1)