The paper considers the finite element approximation of the elliptic interface problem: \(-\nabla (\sigma \nabla u)+cu=f\) in \(\Omega \subset {\mathbb{R}}^ n\), \(n=2\) or 3, with \(u=0\) on \(\partial \Omega\), where \(\sigma\) is discontinuous across a smooth interface \(\Gamma\subset \Omega\). First it is shown that, if the mesh is isoparametrically fitted to \(\Gamma\) using simplicial elements of degree k-1, \(k\geq 2\), then the standard Galerkin method achieves the optimal rate of convergence. Second, since it may be computationally inconvenient to fit the mesh to \(\Gamma\) the authors analyze a piecewise linear approximation of a related penalized problem, as introduced by I. Babuška [Computing 5, 207-213 (1970; Zbl 0199.506)], based on a mesh that is independent of \(\Gamma\). The optimal rate of convergence is shown here, as well.