# Lagrange multiplier and Dirichlet boundary condition **URL:** \ **Category:** General Discussion\ **Created:** [April 1, 2025, 10:41pm UTC](https://community.freefem.org/t/lagrange-multiplier-and-dirichlet-boundary-condition/3829 "2025-04-01T22:41:33Z")\ **Posts on this page:** 5\ **Page:** 1
**Author:** ![Hemanta822](https://yyz2.discourse-cdn.com/flex030/user_avatar/community.freefem.org/hemanta822/32/2596_2.png) [@Hemanta822](https://community.freefem.org/u/Hemanta822)\ **Post date:** [April 1, 2025, 10:41pm UTC](https://community.freefem.org/t/lagrange-multiplier-and-dirichlet-boundary-condition/3829/1 "2025-04-01T22:41:33Z")
Hi all, I am trying to solve poroelasticity in a rectangular domain [0,1]\times[0,1]. The four sides are labeled as 1,2,3,4 and at the top boundary \Gamma (label 4), I have term that includes Lagrange Multiplier as \ and at \Gamma I want to weakly impose u\_p = u\_s, where u\_s is known. I wrote the following code: ```auto problem poroelasticity([u1p,u2p,p,g1,g2], [v1p,v2p,q,t1,t2]) = int2d(Lh) ( 2*nup*((dx(u1p)*dx(v1p)) + 0.5*(dx(u2p) + dy(u1p))*(dx(v2p) + dy(v1p)) + (dy(u2p)*dy(v2p))) // 2nu(D(u),D(v)) + lambdap*(dx(u1p) + dy(u2p))*(dx(v1p) + dy(v2p)) // lambda(div(u),div(v)) - alphap*(p*dx(v1p) + p*dy(v2p)) // alpha(p,div(v)) + (spp/dt)*p*q // s/dt(p,q) + (alphap/dt)*(dx(u1p)*q+ dy(u2p)*q) // alpha/dt(div(u),q) + kappinvp*(dx(p)*dx(q) + dy(p)*dy(q)) //kappa(\grad p, \grad q) ) - int2d(Lh)( f1p*v1p + f2p*v2p + f3p*q + (spp/dt)*pold*q + (alphap/dt)*(dx(u1pold)*q + dy(u2pold)*q) ) + int1d(Lh,4) (g1*v1p + g2*v2p) + int1d(Lh,4) (u1p*t1 + u2p*t2) - int1d(Lh,4) (u1s*t1 + u2s*t2) + on(1,2,3, u1p=u1ptrue, u2p=u2ptrue, p = ptrue);// BC ``` However it keeps giving me the following error message: ```auto kk 2 6 : umfpack_di_numeric failed current line = 263 Exec error : umfpack_di_numeric failed -- number :1 Exec error : umfpack_di_numeric failed -- number :1 ``` I would be grateful if someone could help me fix this issue. Thank you! ---
**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:** [April 2, 2025, 9:05am UTC](https://community.freefem.org/t/lagrange-multiplier-and-dirichlet-boundary-condition/3829/2 "2025-04-02T09:05:24Z")
Dear Hemanta, Your matrix is singular if `g1,g2,t1,t2` are in a fespace over the whole mesh Lh (I don’t have this information from your post), because the value of `g1,g2` is not determined inside the domain. A way to cure this is to add a small term `+int2d(Lh)(1.e-8*(g1*t1+g2*t2))` making the problem well-posed inside the domain, without perturbing your boundary formulation. Otherwise a more clear formulation is to take `g1,g2,t1,t2` defined only on the boundary \Gamma. An example doing this is here > [@Lagrange multiplier for Dirichlet boundary condition](https://community.freefem.org/t/lagrange-multiplier-for-dirichlet-boundary-condition/2402/7): > > Dear @cmd @marchywka , thank you both for your previous help and I have made good process about using Lagrange multiplier to implement Dirichlet BC. However, there are problems when I try to parallelise my code. It seems FreeFemm++ and ff-mpirun -n 1 … compile code differently. I found the fowllowing three operations are not allowed using ff-mpirun -n 1…, while it works well for FreeFem++ 1. int[int] labs = [1]; ThS = extract(ThG, label=labs); 2. matrix UAtemp = ua(Vh,Sh); 3. createMat(ThS… as `stokes_lagrange.edp` There the matrix is built by blocks. Probably you can do also by using a composite space. ---
**Author:** ![Hemanta822](https://yyz2.discourse-cdn.com/flex030/user_avatar/community.freefem.org/hemanta822/32/2596_2.png) [@Hemanta822](https://community.freefem.org/u/Hemanta822)\ **Post date:** [April 2, 2025, 3:47pm UTC](https://community.freefem.org/t/lagrange-multiplier-and-dirichlet-boundary-condition/3829/3 "2025-04-02T15:47:37Z")
Thank you Francois. Yes, I had defined `g1, g2, t1, t2` over the whole mesh `Lh`. Your suggestion of using a small term is helping me now. I had gone through the discussion you mentioned (even before posting my approach here), but I felt very difficult to follow it. I will try to implement using composite space. Thank you again. ---
**Author:** ![Hemanta822](https://yyz2.discourse-cdn.com/flex030/user_avatar/community.freefem.org/hemanta822/32/2596_2.png) [@Hemanta822](https://community.freefem.org/u/Hemanta822)\ **Post date:** [April 4, 2025, 5:39pm UTC](https://community.freefem.org/t/lagrange-multiplier-and-dirichlet-boundary-condition/3829/4 "2025-04-04T17:39:44Z")
Thank you @fb77 , I modified my code. I was able to extract \Gamma, define g in \Gamma and use composite spaces. Your suggestion helped me. Thank you again! ---
**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:** [April 5, 2025, 9:33am UTC](https://community.freefem.org/t/lagrange-multiplier-and-dirichlet-boundary-condition/3829/5 "2025-04-05T09:33:24Z")
Great that it works! 😀