A linear Galerkin numerical method for a quasilinear subdiffusion equation

TL;DR

This paper introduces a spectral Galerkin method combined with L1 discretization for solving quasilinear subdiffusion equations with nonlinear diffusivity and source terms, providing stability, convergence, and improved error constants.

Contribution

It develops a novel numerical scheme coupling spectral Galerkin in space with L1 in time for quasilinear subdiffusion equations, with proven stability and spectral accuracy.

Findings

Proves stability and convergence of the method.

Achieves spectral accuracy in space.

Determines asymptotic exact error constants.

Abstract

We couple the L1 discretization for Caputo derivative in time with spectral Galerkin method in space to devise a scheme that solves quasilinear subdiffusion equations. Both the diffusivity and the source are allowed to be nonlinear functions of the solution. We prove method's stability and convergence with spectral accuracy in space. The temporal order depends on solution's regularity in time. Further, we support our results with numerical simulations that utilize parallelism for spatial discretization. Moreover, as a side result we find asymptotic exact values of error constants along with their remainders for discretizations of Caputo derivative and fractional integrals. These constants are the smallest possible which improves the previously established results from the literature.