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Finite element approximation for time-dependent diffusion with measure-valued source

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Abstract

The convergence of finite element methods for elliptic and parabolic partial differential equations is well-established if source terms are sufficiently smooth. Noting that finite element computation is easily implemented even when the source terms are measure-valued—for instance, modeling point sources by Dirac delta distributions—we prove new convergence order results in two and three dimensions both for elliptic and for parabolic equations with measures as source terms. These analytical results are confirmed by numerical tests using COMSOL Multiphysics.

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Acknowledgments

The authors are indebted to Andreas Prohl for valuable input to the analysis of this paper. The hardware used in the computational studies is part of the UMBC High Performance Computing Facility (HPCF). The facility is supported by the U.S. National Science Foundation through the MRI program (Grant no. CNS-0821258) and the SCREMS program (Grant no. DMS-0821311), with additional substantial support from the University of Maryland, Baltimore County (UMBC). See http://www.umbc.edu/hpcf for more information on HPCF and the projects using its resources. M. Kružík was partially supported by the grants IAA 100750802 (GA AV ČR), P201/10/0357, and P105/11/0411 (GA ČR). D. W. Trott also acknowledges financial support as HPCF RA.

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Authors and Affiliations

  1. Department of Mathematics and Statistics, University of Maryland, Baltimore County, Baltimore, MD, 21250, USA

    Thomas I. Seidman, Matthias K. Gobbert & David W. Trott

  2. Institute of Information Theory and Automation, Academy of Sciences of the Czech Republic, 182 08, Praha 8, Czech Republic

    Martin Kružík

  3. Faculty of Civil Engineering, Czech Technical University, 166 29, Praha 6, Czech Republic

    Martin Kružík

Authors
  1. Thomas I. Seidman
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  2. Matthias K. Gobbert
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  3. David W. Trott
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  4. Martin Kružík
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Correspondence to Thomas I. Seidman.

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Seidman, T.I., Gobbert, M.K., Trott, D.W. et al. Finite element approximation for time-dependent diffusion with measure-valued source. Numer. Math. 122, 709–723 (2012). https://doi.org/10.1007/s00211-012-0474-8

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  • DOI: https://doi.org/10.1007/s00211-012-0474-8

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