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Consider the problem

$$\partial_t u-\Delta u = f\\ u(\Sigma_1)=f_D\\ \partial_\nu u (\Sigma_2)=f_N\\u(0)=u_0$$

where the sides of the space-time cylinder $\Sigma_i$ are disjoint (one of them could be empty, so as to recover a non-mixed problem).

Do you know of any reference where a priori estimates using finite elements in space and some sort of time stepping in time are derived in such generality? Anything even only slightly more complicated than homogeneous Dirichlet boundary conditions is welcome.

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Take a look at the papers by Vidar Thomee. Maybe Jim Bramble also has something. This would have been done in the 1970s and 1980s, maybe going into the 1990s.

You will probably find that the points where you switch from Dirichlet to Neumann conditions, at least if they are not at corners of the domain, produce a corner-type singularity. As a consequence, you shouldn't expect full regularity of the solution to produce optimal error estimates.

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