  We prove quasi-optimal a priori error estimates for finite element
approximations of boundary normal fluxes in the $L^2$-norm. Our results are
valid for a variety of different schemes for weakly enforcing Dirichlet
boundary conditions including Nitsche's method, and Lagrange multiplier
methods. The proof is based on an error representation formula that is derived
by using a discrete dual problem with $L^2$-Dirichlet boundary data and
combines a weighted discrete stability estimate for the dual problem with
anisotropic interpolation estimates in the boundary zone.

Numerical Analysis