Convolution Quadrature for the quasilinear subdiffusion equation

TL;DR

This work develops a Convolution Quadrature scheme, combined with finite element spatial discretization, to solve the quasilinear subdiffusion equation with Caputo time derivative of order $\alpha$. The authors prove unconditional stability and convergence, derive globally optimal time error bounds (and pointwise rates for $\alpha\geq 1/2$), and extend the analysis to the quasilinear setting via an auxiliary linear problem and energy methods. A discrete Grönwall inequality for CQ is established, enabling rigorous error control, while a fast, oblivious implementation reduces memory to $O(\log N)$ and computational complexity to $O(N\log N)$. Numerical experiments confirm the predicted convergence behavior and demonstrate significant time savings, validating the method for both semilinear and quasilinear cases and highlighting practical benefits for time-fractional PDE simulations.

Abstract

We construct a Convolution Quadrature (CQ) scheme for the quasilinear subdiffusion equation of order $α$ 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 $0<α<1$ and pointwise for $α\geq 1/2$ 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 Grönwall 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.

Figures (5)

• Figure 1: Plot of the BDF(2) weights $w_j$ as in \ref{['eqn:BDF2Weights']} for different $0<\alpha<1$ with $h=1$. Weight $w_2$ is negative for $0<\alpha<5/8$, weight $w_3$ is negative for $0<\alpha<7/8$, while all other weights are always negative.
• Figure 2: Plot of the BDF(2) weights $w_j$ as in \ref{['eqn:BDF2Weights']} for different $j\geq 4$ with $h=1$.
• Figure 3: Error $\|u(T) - U^n\|$ with $nh = T = 1$ for the problem \ref{['eqn:NumExampleExact']}. On the left: Euler CQ. On the right: BDF2-CQ.
• Figure 4: Error $\|u(h) - U^1\|$ for the problem \ref{['eqn:NumExampleExact']}. On the left: Euler CQ. On the right: BDF2-CQ.
• Figure 5: Mean ratio of calculation times: without and with the implementation of the fast and oblivious algorithm. The comparison with the L1 scheme is also presented.

Theorems & Definitions (27)

• Proposition 1
• proof
• Remark 1
• Proposition 2
• proof
• Proposition 3: Discrete fractional Grönwall inequality (integral version) (dixon1985order, Theorem 2.1)
• Lemma 1
• proof
• Lemma 2: Discrete Grönwall inequality for the convolution quadrature
• proof