A linearized L2-1[formula omitted] Galerkin FEM for Kirchhoff type quasilinear subdiffusion equation with memory

Abstract

This article aims to construct a novel linearized Galerkin FEM based on Alikhanov’s L2-1σ scheme to solve a Kirchhoff type quasilinear subdiffusion equation with memory Dα. Near time t=0, the weak singularity of the solution to the equation Dα is taken into account by using the graded mesh in the time direction. The nonlocal behavior of Kirchhoff term causes huge computer storage thereby increasing computational cost. We reduce these costs by linearizing the Kirchhoff term. We derive a priori bounds and global rates of convergence for the developed numerical scheme. We prove that the convergence rate of the proposed numerical scheme is first-order in space and second-order in time in L∞(0,T;L2(Ω)). In addition, we prove the same convergence rate in L∞(0,T;H01(Ω)) by making use of a discrete Laplacian operator. A numerical experiment is presented to verify the theoretical results.