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We prove quasi-optimal a priori error estimates for finite elementapproximations of boundary normal fluxes in the $L^2$-norm. Our results arevalid for a variety of different schemes for weakly enforcing Dirichletboundary conditions including Nitsche's method, and Lagrange multipliermethods. The proof is based on an error representation formula that is derivedby using a discrete dual problem with $L^2$-Dirichlet boundary data andcombines a weighted discrete stability estimate for the dual problem withanisotropic interpolation estimates in the boundary zone.

$L^2$-error estimates for finite element approximations of boundary fluxes