# Problem imposing tangent stress condition

**URL:** <https://community.freefem.org/t/problem-imposing-tangent-stress-condition/3065>\
**Category:** General Discussion\
**Created:** [March 25, 2024, 7:08pm UTC](https://community.freefem.org/t/problem-imposing-tangent-stress-condition/3065 "2024-03-25T19:08:54Z")\
**Posts on this page:** 5\
**Page:** 1

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**Author:** ![ilavi](https://avatars.discourse-cdn.com/v4/letter/i/b782af/32.png) [@ilavi](https://community.freefem.org/u/ilavi)\
**Post date:** [March 25, 2024, 7:08pm UTC](https://community.freefem.org/t/problem-imposing-tangent-stress-condition/3065/1 "2024-03-25T19:08:54Z")

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Hi,

I am solving a compressible Stokes problem with a prescribed pressure on a hexagon with the following BCs on each face i:

1. u.ni =0
2. ti.(sigma.ni) = gi

But in my scheme the 2nd condition is not imposed to a satisfactory degree.

As explained [here](https://community.freefem.org/t/normal-boundary-condition/341/3), I am using penalization for the first condition and projection on the tangent to impose the 2nd condition.

```auto
problem stokes([ux,uy],[vx,vy], solver=sparsesolver) = 
        - int2d(Th)( grad(pr)'*[vx, vy] ) // pressure
        - int2d(Th)( eta*Strain(ux, uy)'*Grad(vx, vy) ) // viscosity
        - int2d(Th)( xi*[ux, uy]'*[vx, vy] ) // friction 
        - int1d(Th,1)( g1 * [T1[0], T1[1]]'*[vx, vy] ) // ti.(sigma.ni) = gi
        - int1d(Th,2)( g2 * [T2[0], T2[1]]'*[vx, vy] )    
        - int1d(Th,3)( g3 * [T3[0], T3[1]]'*[vx, vy] )   
        - int1d(Th,4)( g4 * [T4[0], T4[1]]'*[vx, vy] )    
        - int1d(Th,5)( g5 * [T5[0], T5[1]]'*[vx, vy] )     
        - int1d(Th,6)( g6 * [T6[0], T6[1]]'*[vx, vy] )             
        - int1d(Th)( penal * (N'*[ux, uy]) * (N'*[vx, vy]) ); // penalize u.n

```

Note: each gi is defined as P1 on a line with tangent Ti (precomputed). The line meshes coincide with the Hexagon boundaries. The issue persists also if g are defined in 2D, and regardless of interpolation order.

The discrepancy between the actual ti.(sigma.ni) (after solving for u) and the input gi is what I am struggling to fix. The problem is mostly concentrated near the edges of each line but not only.

I appreciate the help. Thanks.

[ffbctest.edp](https://community.freefem.org/uploads/short-url/nr5Vu4koDmhmXO8MEMtGtzxIdCV.edp) (5.3 KB)

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<div class="post-metadata">

**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 26, 2024, 11:27am UTC](https://community.freefem.org/t/problem-imposing-tangent-stress-condition/3065/2 "2024-03-26T11:27:54Z")

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Hi,  
Your code looks correct. I checked that the error in the tangential boundary condition decreases when refining the mesh, linearly in the mesh size. I think that the corners of the domain induce some singularities in the solution, so that the error is not as small as you expect. Nevertheless I think that what you get is correct.

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<div class="post-metadata">

**Author:** ![ilavi](https://avatars.discourse-cdn.com/v4/letter/i/b782af/32.png) [@ilavi](https://community.freefem.org/u/ilavi)\
**Post date:** [March 26, 2024, 2:27pm UTC](https://community.freefem.org/t/problem-imposing-tangent-stress-condition/3065/3 "2024-03-26T14:27:58Z")

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Thank you François,  
Could it be that imposing such BCs through sharp corners is somehow ill-posed?  
Note that a clearer way to see the problem is setting all g’s to 1.

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<div class="post-metadata">

**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 26, 2024, 4:01pm UTC](https://community.freefem.org/t/problem-imposing-tangent-stress-condition/3065/4 "2024-03-26T16:01:13Z")

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I think that the problem is well-posed because the conditions are set a.e. on the boundary (at the continuous level).  
But the solution `u` is probably in the Sobolev space `H^2` and no more than 2 derivatives.  
Looking at the numerical error on the BCs, if you take a test function `v` which is arbitrary on the boundary satifying `v.N=0`, and vanishing for the degreees of freedom inside the domain, you see that the boundary error `E` on the tangent stress verifies `\int_\Gamma E.v=\int_\Omega div(\sigma).v`  
Assume that `div(\sigma)` is in `L^\infty`. Then taking `v=E` on the boundary you get  
that `v` is in `L^4` in `\Omega` (because 2d) estimated by the `L^2` norm of `E` on the boundary. Then `||E||` in `L^2` is of the order of (the measure of the cells of \Omega that touch the boundary)^{3/4}, which is h^3/4 .  
Thus the error is going to zero slowly as h tends to zero.

When you measure the error numerically you have to normalize your estimate by the length of the boundary, which is about 36.  
In the attached file I compute the normalized `L^1` error on each of the 6 boundaries  
[ffbctest.edp](https://community.freefem.org/uploads/short-url/yjksnjPvzGh77j3SAnKwkLVhlAc.edp) (5.7 KB)  
I get the errors  
0.0839251 0.088971 0.0793675 0.00596786 0.00204029 0.0143004  
You see that the normalized measured error is indeed not as bad as you thought!

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<div class="post-metadata">

**Author:** ![ilavi](https://avatars.discourse-cdn.com/v4/letter/i/b782af/32.png) [@ilavi](https://community.freefem.org/u/ilavi)\
**Post date:** [March 26, 2024, 6:19pm UTC](https://community.freefem.org/t/problem-imposing-tangent-stress-condition/3065/5 "2024-03-26T18:19:08Z")

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Thanks very much for this analysis. I will examine the convergence rate and verify that exponent.