TY - JOUR
T1 - Convergence and superconvergence analyses of HDG methods for time fractional diffusion problems
AU - Mustapha, Kassem
AU - Nour, Maher
AU - Cockburn, Bernardo
N1 - Publisher Copyright:
© 2015, Springer Science+Business Media New York.
PY - 2016/4/1
Y1 - 2016/4/1
N2 - We study the hybridizable discontinuous Galerkin (HDG) method for the spatial discretization of time fractional diffusion models with Caputo derivative of order 0 < α < 1. For each time t ∈ [0, T], when the HDG approximations are taken to be piecewise polynomials of degree k ≥ 0 on the spatial domain Ω, the approximations to the exact solution u in the L∞(0, T; L2(Ω))-norm and to ∇u in the (Formula presented.) -norm are proven to converge with the rate hk+1 provided that u is sufficiently regular, where h is the maximum diameter of the elements of the mesh. Moreover, for k ≥ 1, we obtain a superconvergence result which allows us to compute, in an elementwise manner, a new approximation for u converging with a rate hk+2 (ignoring the logarithmic factor), for quasi-uniform spatial meshes. Numerical experiments validating the theoretical results are displayed.
AB - We study the hybridizable discontinuous Galerkin (HDG) method for the spatial discretization of time fractional diffusion models with Caputo derivative of order 0 < α < 1. For each time t ∈ [0, T], when the HDG approximations are taken to be piecewise polynomials of degree k ≥ 0 on the spatial domain Ω, the approximations to the exact solution u in the L∞(0, T; L2(Ω))-norm and to ∇u in the (Formula presented.) -norm are proven to converge with the rate hk+1 provided that u is sufficiently regular, where h is the maximum diameter of the elements of the mesh. Moreover, for k ≥ 1, we obtain a superconvergence result which allows us to compute, in an elementwise manner, a new approximation for u converging with a rate hk+2 (ignoring the logarithmic factor), for quasi-uniform spatial meshes. Numerical experiments validating the theoretical results are displayed.
KW - Anomalous diffusion
KW - Convergence analysis
KW - Discontinuous Galerkin methods
KW - Hybridization
KW - Time fractional
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U2 - 10.1007/s10444-015-9428-x
DO - 10.1007/s10444-015-9428-x
M3 - Article
AN - SCOPUS:84945261363
SN - 1019-7168
VL - 42
SP - 377
EP - 393
JO - Advances in Computational Mathematics
JF - Advances in Computational Mathematics
IS - 2
ER -