Semidiscrete finite element analysis of time fractional parabolic problems: A unified approach

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Abstract

In this paper, we consider the numerical approximation of time-fractional parabolic problems involving Caputo derivatives in time of order a, 0 < a < 1. We derive optimal error estimates for semidiscrete Galerkin finite element (FE) type approximations for problems with smooth and nonsmooth initial data. Our analysis relies on energy arguments and exploits the properties of the inverse of the associated elliptic operator. We present the analysis in a general setting so that it is easily applicable to various spatial approximations such as conforming and nonconforming FE methods (FEMs) and FEM on nonconvex domains. The finite element approximation in mixed form is also presented and new error estimates are established for smooth and nonsmooth initial data. Finally, an extension of our analysis to a multiterm time-fractional model is discussed.

Original languageEnglish
Pages (from-to)1673-1692
Number of pages20
JournalSIAM Journal on Numerical Analysis
Volume56
Issue number3
DOIs
Publication statusPublished - Jan 1 2018

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Keywords

  • Mixed method
  • Multiterm fractional diffusion
  • Nonsmooth initial data
  • Optimal error estimates
  • Semidiscrete finite element scheme
  • Time-fractional parabolic equation

ASJC Scopus subject areas

  • Numerical Analysis

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