Vol. 3, No. 1, 2008

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ISSN: 2157-5452 (e-only)
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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.

J. Thomas Beale, David L. Chopp, Randall J. LeVeque and Zhilin Li

Vol. 3 (2008), No. 1, 95–101
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 2000
Primary: 65N06, 65N30, 65N50
Milestones
Received: 7 July 2008
Accepted: 7 July 2008
Published: 20 September 2008
Authors
J. Thomas Beale
Department of Mathematics
Duke University
Box 90320
Durham, NC 27708
United States
David L. Chopp
Engineering Sciences and Applied Mathematics Dept.
Northwestern University
2145 Sheridan Road
Evanston, IL 60208
United States
Randall J. LeVeque
Department of Applied Mathematics
University of Washington
Seattle, WA 98195-2420
United States
Zhilin Li
Center for Research in Scientific Computation
and Department of Mathematics
North Carolina State University
Raleigh, NC 27695
United States