# How to impose interface condition ∂u⁺/∂n = k ∂u⁻/∂n in FreeFem++
**URL:** https://community.freefem.org/t/how-to-impose-interface-condition-u-n-k-u-n-in-freefem/4084
**Category:** General Discussion
**Created:** [September 30, 2025, 3:48am UTC](https://community.freefem.org/t/how-to-impose-interface-condition-u-n-k-u-n-in-freefem/4084 "2025-09-30T03:48:16Z")
**Posts on this page:** 5
**Page:** 1
### Author:  [@Kamjd](https://community.freefem.org/u/Kamjd)
#### Post date: [September 30, 2025, 3:48am UTC](https://community.freefem.org/t/how-to-impose-interface-condition-u-n-k-u-n-in-freefem/4084/1 "2025-09-30T03:48:16Z")
Hello everyone,
I am solving a PDE on a domain Ω split into two sub-domains Ω⁺ and Ω⁻ separated by an interface Γ.
On Γ I need to impose the interface (flux) condition
[
\frac{\partial u^{+}}{\partial n} = k,\frac{\partial u^{-}}{\partial n}
\quad \text{on } \Gamma,
]
where _k_ is a given constant (or function defined on Γ).
I can mesh the two sub-domains separately (or use a single mesh with a label for the interface), and I can write the weak form of the PDE in each sub-domain.
However, I am not sure how to correctly enforce this interface condition in FreeFEM++.
Could someone provide guidance or a small example showing:
- how to write the weak formulation that incorporates this interface condition,
- or how to implement it using Lagrange multipliers or jump terms in FreeFEM++.
A minimal code example would be greatly appreciated.
Thank you in advance for your help!
---
### Author:  [@frederichecht](https://community.freefem.org/u/frederichecht)
#### Post date: [September 30, 2025, 3:18pm UTC](https://community.freefem.org/t/how-to-impose-interface-condition-u-n-k-u-n-in-freefem/4084/2 "2025-09-30T15:18:25Z")
This is a classical problem, if you do heat equation, this two domain two thermal diffusivity k1,k2
the the boundary condition say the thermal flux is continuous , and the u is continuous and in this case the
$$ k\_1 \Delta u\_1 = f\_1 $$ on domain \\Omaga\_1 + BC on external boundary
$$ k\_2 \Delta u\_2 = f\_2 $$ on domain \\Omaga\_2 + BC on external boundary
and k\_1 \\frac{\\partial u\_1}{\\partial n} = k\_2 \\frac{\\partial u\_2}{\\partial n} on \\Gamma
if you call u the solution on \\Omega = \\Omega1 \\cup \\Omega2
k the discontinuous diffusivity of \\Omega
then the Varf is
find u (u given on \\partial \\Omega = g$)
$$ \forall v \H^1\_0(\Omega), int\_\Omega k \nabla u . \nabla v = int\_ f v$$
---
### Author:  [@Kamjd](https://community.freefem.org/u/Kamjd)
#### Post date: [October 3, 2025, 6:32am UTC](https://community.freefem.org/t/how-to-impose-interface-condition-u-n-k-u-n-in-freefem/4084/3 "2025-10-03T06:32:51Z")
Thank you for your answer.
I’m working on a problem where
[
\Delta u^+ = 0 \quad \text{in } \Omega\_1
]
and
[
\Delta u^- = 0 \quad \text{in } \Omega\_2
]
with (\Omega\_1) being a square domain and (\Omega\_2) an ellipse inside (\Omega\_1).
On the interface (\Gamma = \partial\Omega\_2), I impose
[
u^+ = u^- \quad \text{on } \Gamma , \qquad
k\_1 \frac{\partial u^+}{\partial n} = k\_2 \frac{\partial u^-}{\partial n}.
]
So the solution is continuous across the interface, and the flux is continuous as well.
I tried to reproduce the discontinuity of (k) as suggested in the FreeFem++ documentation, using:
```c++
solve Poisson(u,v,solver=LU) =
int2d(tout)(dx(u)*dx(v) + dy(u)*dy(v))
- int1d(tout)(k*(nx*dx(u) + ny*dy(u))*v)
+ int2d(tout)(f*v)
+ on(1, u=0)
+ on(2, u=g);
```
and I defined (k) as follows:
```c++
// We create the variable k = 1 or -1
fespace V(tout,P0);
V pl = region;
int inside = pl((x1+x0)/2, (y1+y0)/2);
V chi = (inside == region);
V k = (1 - chi) * (-1) + 1 * chi;
```
I think this should be sufficient to achieve a good modeling of the problem.
---
### Author:  [@frederichecht](https://community.freefem.org/u/frederichecht)
#### Post date: [October 3, 2025, 8:56am UTC](https://community.freefem.org/t/how-to-impose-interface-condition-u-n-k-u-n-in-freefem/4084/4 "2025-10-03T08:56:48Z")
[kamjd.edp](https://community.freefem.org/uploads/short-url/17ikIV7tAXdW1wZx48pMoeJIoc.edp) (571 Bytes)
I have build a small example
first a mesh Th of ]-2,2[^2 square with a ellipse inside label 2 on ellipse and label 1 on square.
```cpp
solve Poisson(u,v,solver=LU) =
int2d(Th)(dx(u)*dx(v) + dy(u)*dy(v))
- int1d(Th,2)(k*v)
+ int2d(Th)(f*v)
+ on(1, u=0);
The variationnal formulation is
$$ \forall v \in H^1(\Omega_1), \int_\Omega \nabla u.\nabla v = \int_\Gamma (k_1-k_2) v$$
and $k = k_1-k_2$;
```
---
### Author:  [@Kamjd](https://community.freefem.org/u/Kamjd)
#### Post date: [October 13, 2025, 2:46am UTC](https://community.freefem.org/t/how-to-impose-interface-condition-u-n-k-u-n-in-freefem/4084/5 "2025-10-13T02:46:39Z")
Sorry for the late reply. Thank you very much for your help. I think I found the solution to my problem.