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Mathematics of Computation

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An extrapolative approach to integration over hypersurfaces in the level set framework


Authors: Catherine Kublik and Richard Tsai
Journal: Math. Comp. 87 (2018), 2365-2392
MSC (2010): Primary 65D30, 65M06
DOI: https://doi.org/10.1090/mcom/3282
Published electronically: January 2, 2018
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Abstract: We provide a new approach for computing integrals over hypersurfaces in the level set framework. The method is based on the discretization (via simple Riemann sums) of the classical formulation used in the level set framework, with the choice of specific kernels supported on a tubular neighborhood around the interface to approximate the Dirac delta function. The novelty lies in the choice of kernels, specifically its number of vanishing moments, which enables accurate computations of integrals over a class of closed, continuous, piecewise smooth, curves or surfaces; e.g., curves in two dimensions that contain a finite number of corners. We prove that for smooth interfaces, if the kernel has enough vanishing moments (related to the dimension of the embedding space), the analytical integral formulation coincides exactly with the integral one wishes to calculate. For curves with corners and cusps, the formulation is not exact but we provide an analytical result relating the severity of the corner or cusp with the width of the tubular neighborhood. We show numerical examples demonstrating the capability of the approach, especially for integrating over piecewise smooth interfaces and for computing integrals where the integrand is only Lipschitz continuous or has an integrable singularity.


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Additional Information

Catherine Kublik
Affiliation: Department of Mathematics, University of Dayton, 300 College Park, Dayton, Ohio 45469
Email: ckublik1@udayton.edu

Richard Tsai
Affiliation: Department of Mathematics, KTH Royal Institute of Technology, SE-100 44 Stockholm, Sweden — and — Department of Mathematics and Institute for Computational Engineering and Sciences, University of Texas at Austin, 2515 Speedway, Austin, Texas 78712
Email: ytsai@ices.utexas.edu

DOI: https://doi.org/10.1090/mcom/3282
Received by editor(s): November 3, 2016
Received by editor(s) in revised form: March 6, 2017, and April 4, 2017
Published electronically: January 2, 2018
Additional Notes: The first author was supported by a University of Dayton Research Council Seed Grant.
The second author was supported partially by a National Science Foundation Grant DMS-1318975 and an ARO Grant no. W911NF-12-1-0519. He also thanks National Center for Theoretical Sciences, Taiwan, for hosting his stay at the center where part of the research for this paper was conducted.
Article copyright: © Copyright 2018 American Mathematical Society

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