Abstract

A recent paper by Vaughan, Smith, and Chopp [Comm. App. Math. Comp. Sci. 1 (2006), 207–228] 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.

Keywords

immersed interface method (IIM), elliptic interface problems, finite difference methods, discontinuous coefficients, singular source term, convergence order

Mathematical Subject Classification

Primary: 65N06, 65N30, 65N50

Authors