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Fast evaluation and high accuracy finite element approximation for the time fractional subdiffusion equation
Author(s) -
Ren Jincheng,
Mao Shipeng,
Zhang Jiwei
Publication year - 2018
Publication title -
numerical methods for partial differential equations
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.901
H-Index - 61
eISSN - 1098-2426
pISSN - 0749-159X
DOI - 10.1002/num.22226
Subject(s) - superconvergence , mathematics , discretization , fractional calculus , finite element method , scheme (mathematics) , mathematical analysis , space (punctuation) , variable (mathematics) , derivative (finance) , computer science , physics , thermodynamics , financial economics , economics , operating system
In this article, an efficient algorithm for the evaluation of the Caputo fractional derivative and the superconvergence property of fully discrete finite element approximation for the time fractional subdiffusion equation are considered. First, the space semidiscrete finite element approximation scheme for the constant coefficient problem is derived and supercloseness result is proved. The time discretization is based on the L 1‐type formula, whereas the space discretization is done using, the fully discrete scheme is developed. Under some regularity assumptions, the superconvergence estimate is proposed and analyzed. Then, extension to the case of variable coefficients is also discussed. To reduce the computational cost, the fast evaluation scheme of the Caputo fractional derivative to solve the fractional diffusion equations is designed. Finally, numerical experiments are presented to support the theoretical results.