Fully discrete Galerkin scheme for a semilinear subdiffusion equation with nonsmooth data and time-dependent coefficient

TL;DR

This work tackles the numerical solution of a time-fractional semilinear diffusion problem with time-dependent diffusivity and nonsmooth data by developing a fully discrete Galerkin method that marries the L1 scheme for the Caputo derivative with a spatial Galerkin discretization. The authors establish stability and convergence under weak regularity assumptions on the diffusion coefficient, deriving an error bound of the form $\|u-u_h^n\| \le C\big((\Delta t)^{\alpha}+h^{p}\|\varphi\|_p+h^2 t_n^{-\alpha(2-p)/2}\|\varphi\|_p\big)$, and prove optimal spatial rates alongside global-in-time temporal rates. The analysis hinges on a discrete fractional Grönwall argument and precise extrapolation error estimates for the nonlinear term. Numerical experiments in 1D with both smooth and nonsmooth data validate the predicted convergence rates: spatial order up to $2$ for smooth data and $1$ for nonsmooth data, with temporal convergence governed by the fractional order $\alpha$, even in the presence of time-dependent diffusivity. Overall, the paper provides a robust, fully discrete framework for semilinear subdiffusion with memory effects relevant to applications featuring nonsmooth inputs and evolving media.

Abstract

We couple the L1 discretization of the Caputo fractional derivative in time with the Galerkin scheme to devise a linear numerical method for the semilinear subdiffusion equation. Two important points that we make are: nonsmooth initial data and time-dependent diffusion coefficient. We prove the stability and convergence of the method under weak assumptions concerning regularity of the diffusivity. We find optimal pointwise in space and global in time errors, which are verified with several numerical experiments.

Figures (4)

• Figure 1: A log-log plot of the global in time $L^2$ error as a function of the number of temporal gridpoints $N$ with the number of spatial degrees of freedom $M=2^6$.
• Figure 2: A log-log plot of the local in time $L^2$ error as a function of the number of spatial degrees of freedom $M$ with the number of temporal gridpoints $N=10^3$.
• Figure 3: Exemplary solutions of the time-fractional ZFK equation with variable diffusivity. The solutions are obtained separately, with $N=2^{12}$ and $M=2^6$.
• Figure 4: On the left: solutions of \ref{['eq:ZFK']} at a point $x=1/2$, with $T=10^{-4}$, $N=2^{13}$, $M=2^6+1$. On the right: timer derivatives on a log-log scale.

Theorems & Definitions (7)

• Theorem 1: Stability
• proof
• Lemma 1: Extrapolation error
• proof
• Theorem 2: Convergence
• proof
• Corollary 1