Galerkin FEM for Fractional Order Parabolic Equations with Initial Data in H − s , 0 ≤ s ≤ 1

Bangti Jin; Raytcho Lazarov; Joseph Pasciak; Zhi Zhou

doi:10.1007/978-3-642-41515-9_3

Galerkin FEM for Fractional Order Parabolic Equations with Initial Data in H − s , 0 ≤ s ≤ 1

Bangti Jin, Raytcho Lazarov, Joseph Pasciak, Zhi Zhou

Research output: Chapter in Book/Report/Conference proceeding › Chapter

13 Scopus citations

Abstract

We investigate semi-discrete numerical schemes based on the standard Galerkin and lumped mass Galerkin finite element methods for an initial-boundary value problem for homogeneous fractional diffusion problems with non-smooth initial data. We assume that Ω ⊂ ℝd , d = 1,2,3 is a convex polygonal (polyhedral) domain. We theoretically justify optimal order error estimates in L2- and H1-norms for initial data in H-s (Ω), 0 ≤ s ≤ 1. We confirm our theoretical findings with a number of numerical tests that include initial data v being a Dirac δ-function supported on a (d-1)-dimensional manifold. © 2013 Springer-Verlag.

Original language	English (US)
Title of host publication	Numerical Analysis and Its Applications
Publisher	Springer Nature
Pages	24-37
Number of pages	14
ISBN (Print)	9783642415142
DOIs	https://doi.org/10.1007/978-3-642-41515-9_3
State	Published - 2013
Externally published	Yes

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10.1007/978-3-642-41515-9_3

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title = "Galerkin FEM for Fractional Order Parabolic Equations with Initial Data in H − s , 0 ≤ s ≤ 1",

abstract = "We investigate semi-discrete numerical schemes based on the standard Galerkin and lumped mass Galerkin finite element methods for an initial-boundary value problem for homogeneous fractional diffusion problems with non-smooth initial data. We assume that Ω ⊂ ℝd , d = 1,2,3 is a convex polygonal (polyhedral) domain. We theoretically justify optimal order error estimates in L2- and H1-norms for initial data in H-s (Ω), 0 ≤ s ≤ 1. We confirm our theoretical findings with a number of numerical tests that include initial data v being a Dirac δ-function supported on a (d-1)-dimensional manifold. {\textcopyright} 2013 Springer-Verlag.",

author = "Bangti Jin and Raytcho Lazarov and Joseph Pasciak and Zhi Zhou",

note = "KAUST Repository Item: Exported on 2020-10-01 Acknowledged KAUST grant number(s): KUS-C1-016-04 Acknowledgements: The research of R. Lazarov and Z. Zhou was supportedin parts by US NSF Grant DMS-1016525 and J. Pasciak has been supported byNSF Grant DMS-1216551. The work of all authors has been supported also byAward No. KUS-C1-016-04, made by King Abdullah University of Science andTechnology (KAUST) This publication acknowledges KAUST support, but has no KAUST affiliated authors.",

year = "2013",

doi = "10.1007/978-3-642-41515-9_3",

language = "English (US)",

isbn = "9783642415142",

pages = "24--37",

booktitle = "Numerical Analysis and Its Applications",

publisher = "Springer Nature",

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T1 - Galerkin FEM for Fractional Order Parabolic Equations with Initial Data in H − s , 0 ≤ s ≤ 1

AU - Jin, Bangti

AU - Lazarov, Raytcho

AU - Pasciak, Joseph

AU - Zhou, Zhi

N1 - KAUST Repository Item: Exported on 2020-10-01 Acknowledged KAUST grant number(s): KUS-C1-016-04 Acknowledgements: The research of R. Lazarov and Z. Zhou was supportedin parts by US NSF Grant DMS-1016525 and J. Pasciak has been supported byNSF Grant DMS-1216551. The work of all authors has been supported also byAward No. KUS-C1-016-04, made by King Abdullah University of Science andTechnology (KAUST) This publication acknowledges KAUST support, but has no KAUST affiliated authors.

PY - 2013

Y1 - 2013

N2 - We investigate semi-discrete numerical schemes based on the standard Galerkin and lumped mass Galerkin finite element methods for an initial-boundary value problem for homogeneous fractional diffusion problems with non-smooth initial data. We assume that Ω ⊂ ℝd , d = 1,2,3 is a convex polygonal (polyhedral) domain. We theoretically justify optimal order error estimates in L2- and H1-norms for initial data in H-s (Ω), 0 ≤ s ≤ 1. We confirm our theoretical findings with a number of numerical tests that include initial data v being a Dirac δ-function supported on a (d-1)-dimensional manifold. © 2013 Springer-Verlag.

AB - We investigate semi-discrete numerical schemes based on the standard Galerkin and lumped mass Galerkin finite element methods for an initial-boundary value problem for homogeneous fractional diffusion problems with non-smooth initial data. We assume that Ω ⊂ ℝd , d = 1,2,3 is a convex polygonal (polyhedral) domain. We theoretically justify optimal order error estimates in L2- and H1-norms for initial data in H-s (Ω), 0 ≤ s ≤ 1. We confirm our theoretical findings with a number of numerical tests that include initial data v being a Dirac δ-function supported on a (d-1)-dimensional manifold. © 2013 Springer-Verlag.

UR - http://hdl.handle.net/10754/598393

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SN - 9783642415142

SP - 24

EP - 37

BT - Numerical Analysis and Its Applications

PB - Springer Nature

ER -

Galerkin FEM for Fractional Order Parabolic Equations with Initial Data in H − s , 0 ≤ s ≤ 1

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