Numerical analysis of nonlinear subdiffusion equations

Abstract

We present a general framework for the rigorous numerical analysis of time-fractional nonlinear parabolic partial differential equations, with a fractional derivative of order α ∈ (0, 1) in time. It relies on three technical tools: a fractional version of the discrete Grönwall type inequality, discrete maximal regularity, and regularity theory of nonlinear equations. We establish a general criterion for showing the fractional discrete Grönwall inequality and verify it for the L1 scheme and convolution quadrature generated by backward difference formulas. Further, we provide a complete solution theory, e.g., existence, uniqueness, and regularity, for a time-fractional diffusion equation with a Lipschitz nonlinear source term. Together with the known results of discrete maximal regularity, we derive pointwise L2(Ω) norm error estimates for semidiscrete Galerkin finite element solutions and fully discrete solutions, which are of order O(h2) (up to a logarithmic factor) and O(τα), respectively, without any extra regularity assumption on the solution or compatibility condition on the problem data. The sharpness of the convergence rates is supported by the numerical experiments.