Error Estimates for a Semidiscrete Finite Element Method for Fractional Order Parabolic Equations

Handle URI:
http://hdl.handle.net/10754/597967
Title:
Error Estimates for a Semidiscrete Finite Element Method for Fractional Order Parabolic Equations
Authors:
Jin, Bangti; Lazarov, Raytcho; Zhou, Zhi
Abstract:
We consider the initial boundary value problem for a homogeneous time-fractional diffusion equation with an initial condition ν(x) and a homogeneous Dirichlet boundary condition in a bounded convex polygonal domain Ω. We study two semidiscrete approximation schemes, i.e., the Galerkin finite element method (FEM) and lumped mass Galerkin FEM, using piecewise linear functions. We establish almost optimal with respect to the data regularity error estimates, including the cases of smooth and nonsmooth initial data, i.e., ν ∈ H2(Ω) ∩ H0 1(Ω) and ν ∈ L2(Ω). For the lumped mass method, the optimal L2-norm error estimate is valid only under an additional assumption on the mesh, which in two dimensions is known to be satisfied for symmetric meshes. Finally, we present some numerical results that give insight into the reliability of the theoretical study. © 2013 Society for Industrial and Applied Mathematics.
Citation:
Jin B, Lazarov R, Zhou Z (2013) Error Estimates for a Semidiscrete Finite Element Method for Fractional Order Parabolic Equations. SIAM J Numer Anal 51: 445–466. Available: http://dx.doi.org/10.1137/120873984.
Publisher:
Society for Industrial & Applied Mathematics (SIAM)
Journal:
SIAM Journal on Numerical Analysis
KAUST Grant Number:
KUS-C1-016-04
Issue Date:
Jan-2013
DOI:
10.1137/120873984
Type:
Article
ISSN:
0036-1429; 1095-7170
Sponsors:
The research of R. Lazarov and Z. Zhou was supported in part by US NSF grant DMS-1016525. The work of all authors has been supported also by award KUS-C1-016-04, made by King Abdullah University of Science and Technology (KAUST).
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Publications Acknowledging KAUST Support

Full metadata record

DC FieldValue Language
dc.contributor.authorJin, Bangtien
dc.contributor.authorLazarov, Raytchoen
dc.contributor.authorZhou, Zhien
dc.date.accessioned2016-02-25T13:16:59Zen
dc.date.available2016-02-25T13:16:59Zen
dc.date.issued2013-01en
dc.identifier.citationJin B, Lazarov R, Zhou Z (2013) Error Estimates for a Semidiscrete Finite Element Method for Fractional Order Parabolic Equations. SIAM J Numer Anal 51: 445–466. Available: http://dx.doi.org/10.1137/120873984.en
dc.identifier.issn0036-1429en
dc.identifier.issn1095-7170en
dc.identifier.doi10.1137/120873984en
dc.identifier.urihttp://hdl.handle.net/10754/597967en
dc.description.abstractWe consider the initial boundary value problem for a homogeneous time-fractional diffusion equation with an initial condition ν(x) and a homogeneous Dirichlet boundary condition in a bounded convex polygonal domain Ω. We study two semidiscrete approximation schemes, i.e., the Galerkin finite element method (FEM) and lumped mass Galerkin FEM, using piecewise linear functions. We establish almost optimal with respect to the data regularity error estimates, including the cases of smooth and nonsmooth initial data, i.e., ν ∈ H2(Ω) ∩ H0 1(Ω) and ν ∈ L2(Ω). For the lumped mass method, the optimal L2-norm error estimate is valid only under an additional assumption on the mesh, which in two dimensions is known to be satisfied for symmetric meshes. Finally, we present some numerical results that give insight into the reliability of the theoretical study. © 2013 Society for Industrial and Applied Mathematics.en
dc.description.sponsorshipThe research of R. Lazarov and Z. Zhou was supported in part by US NSF grant DMS-1016525. The work of all authors has been supported also by award KUS-C1-016-04, made by King Abdullah University of Science and Technology (KAUST).en
dc.publisherSociety for Industrial & Applied Mathematics (SIAM)en
dc.subjectFinite element methoden
dc.subjectFractional diffusionen
dc.subjectLumped mass methoden
dc.subjectOptimal error estimatesen
dc.subjectSemidiscrete Gelerkin methoden
dc.titleError Estimates for a Semidiscrete Finite Element Method for Fractional Order Parabolic Equationsen
dc.typeArticleen
dc.identifier.journalSIAM Journal on Numerical Analysisen
dc.contributor.institutionTexas A and M University, College Station, United Statesen
kaust.grant.numberKUS-C1-016-04en
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