Numerical Estimation of a Diffusion Coefficient in Subdiffusion

TL;DR

This paper develops and analyzes a numerical method for estimating a spatially dependent diffusion coefficient in a subdiffusion model involving fractional derivatives, proving convergence and providing numerical validation.

Contribution

It introduces a regularized output least-squares approach with finite element and quadrature discretization, proving convergence and deriving rates under regularity assumptions.

Findings

Proved well-posedness of the continuous inverse problem.

Established convergence of discrete solutions to the continuous solution.

Provided numerical results supporting theoretical convergence rates.

Abstract

In this work, we consider the numerical recovery of a spatially dependent diffusion coefficient in a subdiffusion model from distributed observations. The subdiffusion model involves a Caputo fractional derivative of order $\alpha\in(0,1)$ in time. The numerical estimation is based on the regularized output least-squares formulation, with an $H^1(\Omega)$ penalty. We prove the well-posedness of the continuous formulation, e.g., existence and stability. Next, we develop a fully discrete scheme based on the Galerkin finite element method in space and backward Euler convolution quadrature in time. We prove the subsequential convergence of the sequence of discrete solutions to a solution of the continuous problem as the discretization parameters (mesh size and time step size) tend to zero. Further, under an additional regularity condition on the exact coefficient, we derive convergence rates in a weighted $L^2(\Omega)$ norm for the discrete approximations to the exact coefficient {in the one- and two-dimensional cases}. The analysis relies heavily on suitable nonstandard nonsmooth data error estimates for the direct problem. We provide illustrative numerical results to support the theoretical study.