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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| Published in | Numerische Mathematik Vol. 122; no. 4; pp. 709 - 723 |
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| Main Authors | , , , |
| Format | Journal Article |
| Language | English |
| Published |
Berlin/Heidelberg
Springer-Verlag
01.12.2012
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| Subjects | |
| Online Access | Get full text |
| ISSN | 0029-599X 0945-3245 |
| DOI | 10.1007/s00211-012-0474-8 |
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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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| AbstractList | 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. |
| Author | Trott, David W. Kružík, Martin Seidman, Thomas I. Gobbert, Matthias K. |
| Author_xml | – sequence: 1 givenname: Thomas I. surname: Seidman fullname: Seidman, Thomas I. email: seidman@umbc.edu organization: Department of Mathematics and Statistics, University of Maryland, Baltimore County – sequence: 2 givenname: Matthias K. surname: Gobbert fullname: Gobbert, Matthias K. organization: Department of Mathematics and Statistics, University of Maryland, Baltimore County – sequence: 3 givenname: David W. surname: Trott fullname: Trott, David W. organization: Department of Mathematics and Statistics, University of Maryland, Baltimore County – sequence: 4 givenname: Martin surname: Kružík fullname: Kružík, Martin organization: Institute of Information Theory and Automation, Academy of Sciences of the Czech Republic, Faculty of Civil Engineering, Czech Technical University |
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| Cites_doi | 10.1016/j.cam.2003.12.035 10.1007/978-3-642-65161-8 10.1137/070692261 10.3792/pja/1195521686 10.1007/BF01436386 10.1137/S0036142900377991 10.1007/s00211-006-0033-2 10.1007/BF00281421 10.1007/3-7643-7384-9_28 |
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| References | Lions, Magenes (CR9) 1972 CR4 Hanhart, Gobbert, Izu (CR8) 2004; 169 CR6 Fujiwara (CR3) 1967; 43 CR14 CR13 Gobbert (CR5) 2008; 30 CR12 CR10 Chrysafinos, Hou (CR2) 2002; 40 Scott (CR11) 1973; 21 Grisvard (CR7) 1967; 26 Casado-Díaz, Rebollo, Girault, Mármol, Murat (CR1) 2007; 105 MK Gobbert (474_CR5) 2008; 30 D Fujiwara (474_CR3) 1967; 43 R Scott (474_CR11) 1973; 21 J Casado-Díaz (474_CR1) 2007; 105 P Grisvard (474_CR7) 1967; 26 474_CR14 K Chrysafinos (474_CR2) 2002; 40 474_CR4 474_CR12 474_CR13 474_CR10 AL Hanhart (474_CR8) 2004; 169 474_CR6 JL Lions (474_CR9) 1972 |
| References_xml | – volume: 169 start-page: 431 issue: 2 year: 2004 end-page: 458 ident: CR8 article-title: A memory-efficient finite element method for systems of reaction-diffusion equations with non-smooth forcing publication-title: J. Comput. Appl. Math. doi: 10.1016/j.cam.2003.12.035 – year: 1972 ident: CR9 publication-title: Non-homogeneous Boundary Value Problems and Applications doi: 10.1007/978-3-642-65161-8 – volume: 30 start-page: 2922 issue: 6 year: 2008 end-page: 2947 ident: CR5 article-title: Long-time simulations on high resolution meshes to model calcium waves in a heart cell publication-title: SIAM J. Sci. Comput. doi: 10.1137/070692261 – ident: CR4 – ident: CR14 – ident: CR12 – ident: CR13 – ident: CR10 – volume: 43 start-page: 82 year: 1967 end-page: 86 ident: CR3 article-title: Concrete characterization of the domains of fractional powers of some elliptic differential operators of the second order publication-title: Proc. Japan Acad. doi: 10.3792/pja/1195521686 – ident: CR6 – volume: 21 start-page: 317 year: 1973 end-page: 327 ident: CR11 article-title: Finite element convergence for singular data publication-title: Numer. Math. doi: 10.1007/BF01436386 – volume: 40 start-page: 282 issue: 1 year: 2002 end-page: 306 ident: CR2 article-title: Error estimates for semidiscrete finite element approximations of linear and semilinear parabolic equations under minimal regularity assumptions publication-title: SIAM J. Numer. Anal. doi: 10.1137/S0036142900377991 – volume: 105 start-page: 337 year: 2007 end-page: 374 ident: CR1 article-title: Finite elements approximation of second order linear elliptic equations in divergence form with right-hand side in publication-title: Numer. Math. doi: 10.1007/s00211-006-0033-2 – volume: 26 start-page: 40 year: 1967 end-page: 63 ident: CR7 article-title: Caractérisation de quelques espaces d’interpolation publication-title: Arch. Ration. Mech. Anal. doi: 10.1007/BF00281421 – volume: 169 start-page: 431 issue: 2 year: 2004 ident: 474_CR8 publication-title: J. Comput. Appl. Math. doi: 10.1016/j.cam.2003.12.035 – ident: 474_CR10 – volume: 21 start-page: 317 year: 1973 ident: 474_CR11 publication-title: Numer. Math. doi: 10.1007/BF01436386 – ident: 474_CR4 doi: 10.1007/3-7643-7384-9_28 – volume: 30 start-page: 2922 issue: 6 year: 2008 ident: 474_CR5 publication-title: SIAM J. Sci. Comput. doi: 10.1137/070692261 – volume: 26 start-page: 40 year: 1967 ident: 474_CR7 publication-title: Arch. Ration. Mech. Anal. doi: 10.1007/BF00281421 – volume: 40 start-page: 282 issue: 1 year: 2002 ident: 474_CR2 publication-title: SIAM J. Numer. Anal. doi: 10.1137/S0036142900377991 – ident: 474_CR14 – volume: 105 start-page: 337 year: 2007 ident: 474_CR1 publication-title: Numer. Math. doi: 10.1007/s00211-006-0033-2 – volume: 43 start-page: 82 year: 1967 ident: 474_CR3 publication-title: Proc. Japan Acad. doi: 10.3792/pja/1195521686 – ident: 474_CR12 – ident: 474_CR13 – volume-title: Non-homogeneous Boundary Value Problems and Applications year: 1972 ident: 474_CR9 doi: 10.1007/978-3-642-65161-8 – ident: 474_CR6 |
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| SubjectTerms | Mathematical and Computational Engineering Mathematical and Computational Physics Mathematical Methods in Physics Mathematics Mathematics and Statistics Numerical Analysis Numerical and Computational Physics Simulation Theoretical |
| Title | Finite element approximation for time-dependent diffusion with measure-valued source |
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