  We prove several optimal-order error estimates for a finite-element method
applied to an inhomogeneous Robin boundary value problem (BVP) for the Poisson
equation defined in a smooth bounded domain in $\mathbb{R}^n$, $n=2,3$. The
boundary condition is weakly imposed using Nitsche's method. The Robin BVP is
interpreted as the classical penalty method with the penalty parameter
$\varepsilon$. The optimal choice of the mesh size $h$ relative to
$\varepsilon$ is a non-trivial issue. This paper carefully examines the
dependence of $\varepsilon$ on error estimates. Our error estimates require no
unessential regularity assumptions on the solution. Numerical examples are also
reported to confirm our results.
