Subdiffusion with a time-dependent coefficient: analysis and numerical solution

TL;DR

This paper provides a comprehensive error analysis and numerical solutions for subdiffusion equations with time-dependent coefficients, employing finite element methods and backward Euler convolution quadrature, supported by numerical experiments.

Contribution

It introduces a novel perturbation approach for error analysis of fully discrete solutions with time-dependent coefficients in subdiffusion equations.

Findings

Optimal-order convergence proven for numerical solutions.

Regularity of solutions established for nonsmooth data.

Numerical experiments confirm theoretical results.

Abstract

In this work, a complete error analysis is presented for fully discrete solutions of the subdiffusion equation with a time-dependent diffusion coefficient, obtained by the Galerkin finite element method with conforming piecewise linear finite elements in space and backward Euler convolution quadrature in time. The regularity of the solutions of the subdiffusion model is proved for both nonsmooth initial data and incompatible source term. Optimal-order convergence of the numerical solutions is established using the proven solution regularity and a novel perturbation argument via freezing the diffusion coefficient at a fixed time. The analysis is supported by numerical experiments.