A linear Galerkin numerical method for a quasilinear subdiffusion equation

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.

This is the accepted version of the manuscript published in Applied Numerical Mathematics

References in corpus (4)

• Stochastic modeling in nanoscale biophysics: Subdiffusion within proteins
• Too much regularity may force too much uniqueness
• Linear Galerkin-Legendre spectral scheme for a degenerate nonlinear and nonlocal parabolic equation arising in climatology
• Second order scheme for self-similar solutions of a time-fractional porous medium equation on the half-line

Cited by in corpus (2)

• A note on the L1 discretization error for the Caputo derivative in Hölder spaces
• Fully discrete Galerkin scheme for a semilinear subdiffusion equation with nonsmooth data and time-dependent coefficient