4.1.7. Cauchy boundary condition
A Cauchy boundary condition consist of enforcing a Dirichlet value \(u_\text{D}\) and a Neumann flux \(j_\text{N}\) simultaneously, i.e.
This kind of boundary condition is hard to impose in the finite element method. Enforcing the Dirichlet condition would require for the test function \(v\) to vanish on the boundary. Then, however, any Neumann-type boundary integral contributions \(\langle j_\text{N}, v \rangle\) cannot contribute, since \(v\) is \(0\). Additionally, for e.g. the 1d Poisson equation discussed here, one can impose in total two boundary conditions. If e.g. a Cauchy boundary condition is imposed on the left side, the right side cannot have any boundary conditions. However, omitting the specification of a boundary condition in finite elements automatically leads to a zero-flux Neumann boundary condition, as discussed earlier. This would be in total three boundary conditions, more than possible to impose. In principle, one can however enforce a Cauchy boundary condition by enforcing the value strongly, i.e. with a DirichletBC, add a custom InterfaceEquations class that monitors the slope and adjusts the opposite boundary condition accordingly, i.e. via a Lagrange multiplier that is defined in an ODEEquations helper class. Effectively, this would be a kind of a shooting method. However, this is not discussed here.