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

TL;DR

This paper introduces a linearized fractional Crank-Nicolson finite element method for solving a Kirchhoff type quasilinear subdiffusion equation with memory, accounting for solution singularities at initial time and providing proven convergence rates.

Contribution

It develops a novel linearized fractional Crank-Nicolson FEM that handles solution singularities and proves its convergence for a complex time-fractional integro-differential equation.

Findings

Achieves an $O(M^{-1}+N^{-2})$ convergence rate in $L^{inity}(0,T;L^{2}(\Omega))$ and $L^{inity}(0,T;H^{1}_{0}(\Omega))$ norms.

Numerical experiments confirm the theoretical convergence rates.

Addresses solution singularity at $t=0$ in the error analysis.

Abstract

In this paper, we develop a linearized fractional Crank-Nicolson-Galerkin FEM for Kirchhoff type quasilinear time-fractional integro-differential equation $\left(\mathcal{D}^{\alpha}\right)$. 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 $L^{\infty}(0,T;H^{1}_{0}(\Omega))$, 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.