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I am studying heat equation on a 1-D bar. We now that Neumann conditions at both ends leads to a singular matrix (for finite element methods) in equilibrium. Adding an initial condition can lead to unique solution.

It turns out that if we add a Dirichlet boundary condition in equilibrium at one end, we can still have a unique solution (at least when $u_x(0)=u_x(1)=u(1)=0$ and $\int_0^1f(x)dx=0$).

My question is: does it make sense to have both insulated boundary and prescribed temperature at the same end in the real world? How can one simultaneously maintain insulation and constant temperature?

This question might be asked in physics SE, but since I met this problem from a math textbook, I decide to ask it here.

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applying Dirichlet boundary conditions will override your Neumann boundary conditions in the case of the finite element method (I give this as an example, as you mentioned FEM in your question). Dirichlet and Neumann boundaries should not overlap.

When you impose a prescribed temperature, you are also imposing a heat flux, although implicitly. As you can imagine, that heat flux is directly related to the prescribed temperature and the temperature inside your bar. Given a prescribed temperature, you have a prescribed heat flux (one-to-one relation).

So, to answer your question, in the real world, it does not make sense to impose both, as they are not independent one from another.

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