Galerkin Type Methods for Semilinear Time-Fractional Diffusion Problems

Abstract

We derive optimal \(L^2\)-error estimates for semilinear time-fractional subdiffusion problems involving Caputo derivatives in time of order \(\alpha \in (0,1)\), for cases with smooth and nonsmooth initial data. A general framework is introduced allowing a unified error analysis of Galerkin type space approximation methods. The analysis is based on a semigroup type approach and exploits the properties of the inverse of the associated elliptic operator. Completely discrete schemes are analyzed in the same framework using a backward Euler convolution quadrature method in time. Numerical examples including conforming, nonconforming and mixed finite element methods are presented to illustrate the theoretical results.

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Correspondence to Samir Karaa.

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This research was supported by the Research Council of Oman Grant ORG/CBS/15/001.

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Karaa, S. Galerkin Type Methods for Semilinear Time-Fractional Diffusion Problems. J Sci Comput 83, 46 (2020). https://doi.org/10.1007/s10915-020-01230-z

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Keywords

  • Semilinear fractional diffusion
  • Galerkin method
  • Nonconforming FE method
  • Mixed FE method
  • Convolution quadrature
  • Error estimate

Mathematics Subject Classification

  • 65M60
  • 65M12
  • 65M15