PDE coupling on two meshes

Hello,

I am studying the coupling of the bidomain equations (on a 2D finite element mesh ThB corresponding ) with a Laplace equation (on a 2D finite element mesh ThL). However I am faced with the following boundary conditions :

• Dirichlet BC : on the common edge between ThB and ThL I have uB = uL. Where uB is one of the two unkwown the bidomain equation and uL the unknown in the Laplace equation. However, the common edge is not conformal. Should I do an interpolation on this border ? I cheated by considering a little overlapping which leads to a much more satisfactory approximation.

• Neumann BC: on the same common edge, I have \nabla uB \cdot nB = -\nabla uL \cdot nL where nL (resp. nB) is the unit vector normal to ThL (resp. ThB). I then considered :

 int1d(ThB,neumannBC) ((dx(uL)*N.x+dy(uL)*N.y)*vh)


in the bidomain problem and

    int1d(ThL,neumannBC)((dx(uB)*N.x+dy(uB)*N.y)*vh)


for the Laplace equation.
However, it seems to not work correctly. Did I miss something in the consideration of the Neumann conditions?

Thanks a lot !