DISC : Which boundary conditions to solve?

[KikeM]

Implementation details with the boundary conditions arise when the basis is built.
Ideally, the basis should be built for homogeneous/natural boundary conditions.

• An "easy" workaround is to solve the heat equation for natural boundary conditions du/dn = 0 for the whole domain (although I am not sure if existence is guaranteed).
• Solve for homogeneous Dirichlet boundary conditions for the moment.
  There is no strict need to parametrize the boundary conditions for our purposes.
• The generic implementation is to introduce a lifting function.

[KikeM]

Indeed, what we do is to introduce a lifting of the Dirichlet boundary conditions.

If we are in the one-dimensional domain the lifting is trivial, since a spatial interpolation of the boundary conditions is enough.

However, if we are in higher dimensional settings, a smooth interpolation needs to be obtained via the laplacian equation for instance.
According to how the specific boundary conditions of the problem are, we might or might not be able to reduce the lifting extension too.