Correction to the article “A comparison of the extended finite element method with the immersed interface method for elliptic equations with discontinuous coefficients and singular sources” by Vaughan et al.

*(English)*Zbl 1175.65134Summary: A recent paper by B. Vaughan, B. G. Smith and D. L. Chopp [Commun. Appl. Math. Comput. Sci. 1, 207–228 (2006; Zbl 1153.65373)] reported numerical results for three examples using the immersed interface method (IIM) and the extended finite element method (X-FEM). The results presented for the IIM showed first-order accuracy for the solution and inaccurate values of the normal derivative at the interface. This was due to an error in the implementation.

The purpose of this note is to present correct results using the IIM for the same examples used in that paper, which demonstrate the expected second-order accuracy in the maximum norm over all grid points. Results now indicate that on these problems the IIM and X-FEM methods give comparable accuracy in solution values. With appropriate interpolation it is also possible to obtain nearly second order accurate values of the solution and normal derivative at the interface with the IIM.

The purpose of this note is to present correct results using the IIM for the same examples used in that paper, which demonstrate the expected second-order accuracy in the maximum norm over all grid points. Results now indicate that on these problems the IIM and X-FEM methods give comparable accuracy in solution values. With appropriate interpolation it is also possible to obtain nearly second order accurate values of the solution and normal derivative at the interface with the IIM.

##### MSC:

65N30 | Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs |

65N06 | Finite difference methods for boundary value problems involving PDEs |

65N50 | Mesh generation, refinement, and adaptive methods for boundary value problems involving PDEs |

35J25 | Boundary value problems for second-order elliptic equations |

35R05 | PDEs with low regular coefficients and/or low regular data |

65N12 | Stability and convergence of numerical methods for boundary value problems involving PDEs |