Article

Numerische Mathematik

, Volume 122, Issue 4, pp 709-723

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

  • Thomas I. SeidmanAffiliated withDepartment of Mathematics and Statistics, University of Maryland, Baltimore County Email author 
  • , Matthias K. GobbertAffiliated withDepartment of Mathematics and Statistics, University of Maryland, Baltimore County
  • , David W. TrottAffiliated withDepartment of Mathematics and Statistics, University of Maryland, Baltimore County
  • , Martin KružíkAffiliated withInstitute of Information Theory and Automation, Academy of Sciences of the Czech RepublicFaculty of Civil Engineering, Czech Technical University

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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.

Mathematics Subject Classification (2000)

35K57 65M15 65M60