# A priori estimates in finite elements for inhomogeneous heat equation

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.