Error analysis of the space-time interface-fitted finite element method for an inverse source problem for an advection-diffusion equation with moving subdomains

TL;DR

This paper develops and analyzes a space-time interface-fitted finite element method for an inverse source problem in advection-diffusion equations with moving subdomains, providing optimal error estimates and convergence results.

Contribution

It introduces a novel space-time interface-fitted discretization approach with rigorous error analysis for inverse problems involving moving subdomains.

Findings

Achieved optimal error estimates for the discretization methods.

Demonstrated superlinear convergence for the post-processing strategy.

Proposed regularization parameter choices ensuring strong convergence.

Abstract

A space-time interface-fitted approximation of an inverse source problem for the advection-diffusion equation with moving subdomains is investigated. The problem is reformulated as an optimization problem using Tikhonov regularization. A space-time interface-fitted method is employed to discretize the advection-diffusion equation, where two second-order a priori error estimates are established with respect to the $L^2$-norms. Additionally, the regularized source is discretized sequentially using the variational approach, the element-wise constant discretization, and finally, the post-processing strategy. Optimal error estimates are achieved for the first two methods, while superlinear convergence is obtained for the third. Furthermore, a priori choices for the regularization parameter are proposed, depending on the mesh size and noise level. These choices ensure that the discrete and post-processing solutions strongly converge to the exact source as the mesh size and noise level tend to zero.