Convolution Quadrature for the quasilinear subdiffusion equation

Abstract: We construct a Convolution Quadrature (CQ) scheme for the quasilinear subdiffusion equation of order $\alpha$ and supply it with the fast and oblivious implementation. In particular, we find a condition for the CQ to be admissible and discretize the spatial part of the equation with the Finite Element Method. We prove the unconditional stability and convergence of the scheme and find a bound on the error. Our estimates are globally optimal for all $0<\alpha<1$ and pointwise for $\alpha\geq 1/2$ in the sense that they reduce to the well-known results for the linear equation. For the semilinear case, our estimates are optimal both globally and locally. As a passing result, we also obtain a discrete Gr\"onwall inequality for the CQ, which is a crucial ingredient in our convergence proof based on the energy method. The paper is concluded with numerical examples verifying convergence and computation time reduction when using fast and oblivious quadrature.