Numerische Mathematik

, Volume 122, Issue 4, pp 709–723 | Cite as

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

  • Thomas I. Seidman
  • Matthias K. Gobbert
  • David W. Trott
  • Martin Kružík


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.

Mathematics Subject Classification (2000)

35K57 65M15 65M60 



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 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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Copyright information

© Springer-Verlag 2012

Authors and Affiliations

  • Thomas I. Seidman
    • 1
  • Matthias K. Gobbert
    • 1
  • David W. Trott
    • 1
  • Martin Kružík
    • 2
    • 3
  1. 1.Department of Mathematics and StatisticsUniversity of Maryland, Baltimore CountyBaltimoreUSA
  2. 2.Institute of Information Theory and AutomationAcademy of Sciences of the Czech RepublicPraha 8Czech Republic
  3. 3.Faculty of Civil EngineeringCzech Technical UniversityPraha 6Czech Republic

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