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

Bangti Jin, Raytcho Lazarov, Zhi Zhou

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184 Scopus citations

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.
Original languageEnglish (US)
Pages (from-to)445-466
Number of pages22
JournalSIAM Journal on Numerical Analysis
Volume51
Issue number1
DOIs
StatePublished - Jan 2013
Externally publishedYes

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