Fractional Crank-Nicolson-Galerkin Finite Element Scheme for the\n Time-Fractional Nonlinear Diffusion Equation

2018/11/20 by Dileep Kumar, Kumar, Dileep, Sudhakar Chaudhary +3
Engineering · Mathematics · #35R11 #65M12 #65M60 #Analysis of PDEs (math.AP) #Differential Equations and Numerical Methods #FOS: Mathematics #Fractional Differential Equations Solutions #Numerical methods in engineering

paper · pdf · doi:10.48550/arxiv.1811.08485

openalex publication_date 2018/11/20 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

This article presents a finite element scheme with Newton's method for\nsolving the time-fractional nonlinear diffusion equation. For time\ndiscretization, we use the fractional Crank-Nicolson scheme based on backward\nEuler convolution quadrature. We discuss the existence-uniqueness results for\nthe fully-discrete problem. Discrete fractional Gronwall type inequality for\nthe backward Euler convolution quadrature which is used to approximate the\nRiemann-Liouville fractional derivative is established by using the idea of\nD.Li et al. given in [11]. A priori error estimate for the fully-discrete\nproblem in L²(\Ω) norm is derived. Numerical results based on finite\nelement scheme are provided to validate the theoretical estimates.\n

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