# Problem related to SUPG

**URL:** <https://community.freefem.org/t/problem-related-to-supg/3026>\
**Category:** General Discussion\
**Created:** [March 9, 2024, 11:07am UTC](https://community.freefem.org/t/problem-related-to-supg/3026 "2024-03-09T11:07:38Z")\
**Posts on this page:** 8\
**Page:** 1

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**Author:** ![FilippoVaselli](https://avatars.discourse-cdn.com/v4/letter/f/bc79bd/32.png) [@FilippoVaselli](https://community.freefem.org/u/FilippoVaselli)\
**Post date:** [March 9, 2024, 11:07am UTC](https://community.freefem.org/t/problem-related-to-supg/3026/1 "2024-03-09T11:07:38Z")

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I am trying to solve the Navier Stokes equation in a driven cavity but i am acctually having problem with adding the stabilization term related to SUPG.  
Freefam can only solve weak formulation problem and in the integral related to the SUPG terms we are adding the strond formulation of the problem:

fespace Uh(Th,P2);  
Uh ux,uu;  
Uh uy,vv;

fespace Ph(Th,P1);  
Ph p,pp;

//stabilization term related to SUPG  
int2d(Th)(delta_tau_(ux_dx(uu)+uy_dy(vv))_(ux_dx(ux)…)

The problem is related to the term: ux\*dx(ux)  
Can someone help me or does someone have some example to look at?

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**Author:** ![frederichecht](https://yyz2.discourse-cdn.com/flex030/user_avatar/community.freefem.org/frederichecht/32/15_2.png) [@frederichecht](https://community.freefem.org/u/frederichecht)\
**Post date:** [March 10, 2024, 8:16pm UTC](https://community.freefem.org/t/problem-related-to-supg/3026/2 "2024-03-10T20:16:53Z")

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Your stabilization term is non linear so need a schema to solve this problem.

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<div class="post-metadata">

**Author:** ![FilippoVaselli](https://avatars.discourse-cdn.com/v4/letter/f/bc79bd/32.png) [@FilippoVaselli](https://community.freefem.org/u/FilippoVaselli)\
**Post date:** [March 11, 2024, 6:26pm UTC](https://community.freefem.org/t/problem-related-to-supg/3026/3 "2024-03-11T18:26:59Z")

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Thanks a lot for the answer, i’m am new to freefem and i didn’t understand that non linear terms are a problem for freefem. Can i ask you one more thing?

After running a first simulation (without stabilization) with lower reynolds i tried to run a second one with higher Reynolds, so i can use the first one as initial condition for the second. I’m having problem with the SUPG stabilization terms. This term is non linear, but i’am trying some escamotage to solve the problem without linearizing. (driven cavity problem).

Do you have any advice or raccomandation (or see any errors) because the code does not converge.

 ![Immagine WhatsApp 2024-03-09 ore 19.38.27_9f20ac4d](https://canada1.discourse-cdn.com/flex030/uploads/freefem/original/2X/e/ebf4c04f22ffd1f26da546472512969e36e9329e.jpeg)

fespace Uh(Th,P2)  
Uh ux, uu;  
Uh uy,vv;

fespace Ph(Th,P1)  
Ph p,pp;

unastro=1; //reference velocity  
cout\<\<“Reynolds =”\<\<Re\<\<endl;

//SUPG PARAMETERS  
real delta=1; //parametro libero  
Uh U=sqrt(ux^2+uy^2); //modulo velocità  
Uh Reu=U\* hTriangle/(2\*nu); //ReynoldsU  
Uh tau; //parametro di stabilizzazione

tau=(Reu\<3)\* hTriangle\* hTriangle/(12_nu)+(Reu\>=3)_ hTriangle/(2\*U);

Uh pold;

macro div(u1,u2) (dx(u1)+dy(u2)) //  
macro laplaciano(u1) (dxx(u1)+dyy(u1)) //  
macro convettivo(u1,u2,u3) (u1_dx(u3)+u2_dy(u3)) //  
macro prodint(u1,u2,v1,v2) (dx(u1)\*dx(v1)+dy(u2)_dy(v2)+0.5_(dx(u2)\*dx(v2)+  
dx(u2)\*dy(v1)+dy(u1)\*dx(v2)+dy(u1)\*dy(v1))) //

problem NavierStokesSUPG ([ux,uy,p],[uu,vv,pp],init=i) =  
int2d(Th)(alpha\* (ux_uu + uy_vv)  
+ 2\* invRe\* prodint(ux,uy,uu,vv)  
- p\* pp\* (0.000001) - p\* div(uu,vv)  
+ pp\* div(ux,uy))  
- int2d(Th)(alpha\* (uoldx\* uu + uoldy\* vv))  
+ int2d(Th)(convettivo(uoldx,uoldy,ux)\* uu //termini convettivi trattati  
+ convettivo(uoldx,uoldy,uy)\* vv) //in modo semi-implicito  
// Stabilizzazione SUPG  
+ int2d(Th)(delta\* tau\* convettivo(uoldx,uoldy,uu)\* alpha\* ux)  
+ int2d(Th)(delta\* tau\* convettivo(uoldx,uoldy,vv)\* alpha\* uy)  
- int2d(Th)(delta\* tau\* convettivo(ux,uy,uu)_(alpha_ uoldx+dx(pold)-invRe\* laplaciano(uoldx)))  
- int2d(Th)(delta\* tau\* convettivo(ux,uy,vv)_(alpha_ uoldy+dy(pold)-invRe\* laplaciano(uoldy)))  
+ int2d(Th)(delta\* tau\* convettivo(uoldx,uoldy,uu)\* convettivo(uoldx,uoldy,ux))  
+ int2d(Th)(delta\* tau\* convettivo(uoldx,uoldy,vv)\* convettivo(uoldx,uoldy,uy))  
+ on(1,2,4,ux=0,uy=0) + on(3,ux=unastro,uy=0);

pold=p;  
uoldx=ux; //simulazione precedente presa come condizione iniziale  
uoldy=uy;  
NavierStokesSUPG;  
Th=adaptmesh(Th,[ux,uy]);  
plot(Th, wait=1);  
plot([ux,uy], wait=1);

for(i=0; i\<100; i++){  
uoldx=ux;  
uoldy=uy;  
NavierStokesSUPG;  
if (!(i % 2)) // plot every 2 iteration  
plot(p,[ux,uy]);}

plot(p, wait=1);  
plot([ux,uy], wait=1, value=true);

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<div class="post-metadata">

**Author:** ![Walid](https://yyz2.discourse-cdn.com/flex030/user_avatar/community.freefem.org/walid/32/2477_2.png) [@Walid](https://community.freefem.org/u/Walid)\
**Post date:** [June 23, 2024, 2:25am UTC](https://community.freefem.org/t/problem-related-to-supg/3026/4 "2024-06-23T02:25:18Z")

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Hello Did you solve your problem ?  
and if you have a Simple reference regarding SUPG implementation, could you please share it

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**Author:** ![clthu](https://avatars.discourse-cdn.com/v4/letter/c/f04885/32.png) [@clthu](https://community.freefem.org/u/clthu)\
**Post date:** [June 24, 2024, 4:02am UTC](https://community.freefem.org/t/problem-related-to-supg/3026/5 "2024-06-24T04:02:06Z")

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I remember there are some scripts in Stabfem project with SUPG methods to solve the steady solution where the stabilization term is linearized for the jacobian matrix. For the time stepping problem, I think maybe the stabilization term could be added to the source terms in some senses.

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**Author:** ![Walid](https://yyz2.discourse-cdn.com/flex030/user_avatar/community.freefem.org/walid/32/2477_2.png) [@Walid](https://community.freefem.org/u/Walid)\
**Post date:** [June 29, 2024, 11:55pm UTC](https://community.freefem.org/t/problem-related-to-supg/3026/6 "2024-06-29T23:55:52Z")

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Thank you for you reply  
but I really did not understand any thing from stabfem project  
any recommendations for simpler explanation

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<div class="post-metadata">

**Author:** ![Walid](https://yyz2.discourse-cdn.com/flex030/user_avatar/community.freefem.org/walid/32/2477_2.png) [@Walid](https://community.freefem.org/u/Walid)\
**Post date:** [June 30, 2024, 12:00am UTC](https://community.freefem.org/t/problem-related-to-supg/3026/7 "2024-06-30T00:00:06Z")

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could you please Professor explain more about your reply pr suggest a reference

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<div class="post-metadata">

**Author:** ![clthu](https://avatars.discourse-cdn.com/v4/letter/c/f04885/32.png) [@clthu](https://community.freefem.org/u/clthu)\
**Post date:** [July 3, 2024, 6:11am UTC](https://community.freefem.org/t/problem-related-to-supg/3026/8 "2024-07-03T06:11:41Z")

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Hi, there is one example with SUPG to solve the Navier-Stokes equations in stabfem project. They introduce one stabilization term and linearize it in the Newton iteration. Maybe this could give you some ideas to write your own code.

> <https://gitlab.com/stabfem/StabFem/-/blob/master/SOURCES_FREEFEM/Newton_2D_SUPG.edp?ref_type=heads>