Fractional Crank–Nicolson–Galerkin finite element scheme for the time‐fractional nonlinear diffusion equation

This article presents a finite element scheme with Newton's method for solving the time‐fractional nonlinear diffusion equation. For time discretization, we use the fractional Crank–Nicolson scheme based on backward Euler convolution quadrature. We discuss the existence‐uniqueness results for t...

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Bibliographic Details
Published inNumerical methods for partial differential equations Vol. 35; no. 6; pp. 2056 - 2075
Main Authors Kumar, Dileep, Chaudhary, Sudhakar, Srinivas Kumar, V.V.K.
Format Journal Article
LanguageEnglish
Published Hoboken, USA John Wiley & Sons, Inc 01.11.2019
Wiley Subscription Services, Inc
Subjects
Convolution
discrete fractional Gronwall type inequality
error estimates
Finite element method
fractional Crank–Nicolson method
Galerkin method
Newton methods
Quadratures
time‐fractional diffusion equation
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Summary:This article presents a finite element scheme with Newton's method for solving the time‐fractional nonlinear diffusion equation. For time discretization, we use the fractional Crank–Nicolson scheme based on backward Euler convolution quadrature. We discuss the existence‐uniqueness results for the fully discrete problem. A new discrete fractional Gronwall type inequality for the backward Euler convolution quadrature is established. A priori error estimate for the fully discrete problem in L2(Ω) norm is derived. Numerical results based on finite element scheme are provided to validate theoretical estimates on time‐fractional nonlinear Fisher equation and Huxley equation.
ISSN:0749-159X
1098-2426
DOI:10.1002/num.22399