Error Estimates for a Semidiscrete Finite Element Method for Fractional Order Parabolic Equations
KAUST Grant NumberKUS-C1-016-04
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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.
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.
SponsorsThe 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).