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

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 sou...

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Bibliographic Details
Published inNumerische Mathematik Vol. 122; no. 4; pp. 709 - 723
Main Authors Seidman, Thomas I., Gobbert, Matthias K., Trott, David W., Kružík, Martin
Format Journal Article
LanguageEnglish
Published Berlin/Heidelberg Springer-Verlag 01.12.2012
Subjects
Mathematical and Computational Engineering
Mathematical and Computational Physics
Mathematical Methods in Physics
Mathematics
Mathematics and Statistics
Numerical Analysis
Numerical and Computational Physics
Simulation
Theoretical
65M15
35K57
65M60
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Summary: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.
ISSN:0029-599X
0945-3245
DOI:10.1007/s00211-012-0474-8