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I am trying to understand Dirichlet and Neumann boundary conditions in FEM and I wanted to know if my inference is correct. To articulate my understanding, lets consider a simple case of TE and TM propagation in rectangular waveguides.

TE and TM modes

In the case of TE ($E_z$=0 and $H_z$=$\phi$) and in the case of TM ($H_z$=0 and $E_z$=$\phi$). Consequently, in TE we solve for $H_z$ using Helmholtz equation with Neumann boundary condition on all sides and in the case of TM we solve for $E_z$ with Dirichlet boundary condition on all sides.

FEM in Sadiku's book enter image description here

The particular method of FEM and assembly procedure I am using is given by Clayton R Paul and Sadiku wherein the nodal value vector is bifurcated into two sub vectors namely the free node vector($ \phi_{f} $) and the prescribed node vector($ \phi_{p}$). The prescribed node vector is filled by applying Dirichlet boundary conditions to exterior nodes or by assigning excitation to nodes. The free node vector is unknown and has to be computed.

enter image description here

My question is particularly about Neumann boundary condition. My understanding is that since Neumann is implicit and a natural boundary condition in FEM, the node vector($ \phi$) is entirely comprised of free node vector implying that every node in the domain is free and unknown. Is my understanding correct?

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    $\begingroup$ Yes, mostly correct. But Neumann is not always a natural boundary condition: it depends on the problem. Sometimes Dirichlet is natural and implicit (see for example mixed Poisson problem). $\endgroup$
    – knl
    Dec 28, 2017 at 19:12
  • $\begingroup$ I suggest you start from the variational form in order to understand the difference between Dirichlet and Neumann boundary conditions - see this post for some details. $\endgroup$
    – NameRakes
    Dec 31, 2017 at 17:24

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