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On the accuracy of finite element approximations to a class of interface problems


Authors: Johnny Guzmán, Manuel A. Sánchez and Marcus Sarkis
Journal: Math. Comp. 85 (2016), 2071-2098
MSC (2010): Primary 65N30, 65N15
DOI: https://doi.org/10.1090/mcom3051
Published electronically: November 10, 2015
MathSciNet review: 3511275
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Abstract: We define piecewise linear and continuous finite element methods for a class of interface problems in two dimensions. Correction terms are added to the right-hand side of the natural method to render it second-order accurate. We prove that the method is second-order accurate on general quasi-uniform meshes at the nodal points. Finally, we show that the natural method, although non-optimal near the interface, is optimal for points $ \mathcal {O}(\sqrt {h \log (\frac {1}{h})})$ away from the interface.


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Additional Information

Johnny Guzmán
Affiliation: Division of Applied Mathematics, Brown University, Providence, Rhode Island 02912
Email: johnny_guzman@brown.edu

Manuel A. Sánchez
Affiliation: Division of Applied Mathematics, Brown University, Providence, Rhode Island 02912
Email: manuel_sanchez_uribe@brown.edu

Marcus Sarkis
Affiliation: Department of Mathematical Sciences, Worcester Polytechnic Institute, 100 Institute Road, Worcester, Massachusetts 01609
Email: msarkis@wpi.edu

DOI: https://doi.org/10.1090/mcom3051
Keywords: Interface problems, finite elements, pointwise estimates
Received by editor(s): March 28, 2014
Received by editor(s) in revised form: October 10, 2014, December 31, 2014, and February 25, 2015
Published electronically: November 10, 2015
Article copyright: © Copyright 2015 American Mathematical Society