Nitsche's method for a Robin boundary value problem in a smooth domain

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.