Error Estimates for a Linearized Fractional Crank-Nicolson FEM for Kirchhoff type Quasilinear Subdiffusion Equation with Memory

arXiv2208.11104MaRDI QIDQ6408606

Lalit Kumar

Publication date: 23 August 2022

Abstract: In this paper, we develop a linearized fractional Crank-Nicolson-Galerkin FEM for Kirchhoff type quasilinear time-fractional integro-differential equation $left(mathcal{D}^{alpha}ight)$. In general, the solutions to the time-fractional problems exhibit a weak singularity at time $t=0$. This singular behavior of the solutions is taken into account while deriving the convergence estimates of the developed numerical scheme. We prove that the proposed numerical scheme has an accuracy rate of $O(M^{-1}+N^{-2})$ in $L^{infty}(0,T;L^2(Omega))$ as well as in $L^{infty}(0,T;H_0^1(Omega))$, where $M$ and $N$ are the degrees of freedom in the space and time directions respectively. A numerical experiment is presented to verify the theoretical results.

Mathematics Subject Classification ID

Fractional derivatives and integrals (26A33) Functional-differential equations in abstract spaces (34K30) Numerical methods for integral transforms (65R10) Anomalous diffusion models (subdiffusion, superdiffusion, continuous-time random walks, etc.) (60K50)

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