Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

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

Thi Thanh Mai Ta, Quang Huy Nguyen, Dinh Nho Hào

TL;DR

This work develops a space-time interface-fitted finite element framework for an inverse source problem in an advection-diffusion equation with moving subdomains, regularized by Tikhonov regularization. It proves two second-order $L^2$-error estimates for the state and adjoint, and analyzes three discretization strategies for the regularized source: variational, element-wise constant, and post-processing, establishing optimal convergence for the first two and a superlinear rate for the post-processing. The paper also provides a priori choices for the regularization parameter $\lambda$ (balancing mesh size $h$ and data noise $\varepsilon$) to guarantee strong convergence of computed sources to the exact one as $h,\varepsilon \to 0$. The results extend space-time interface techniques to inverse problems with moving interfaces and yield explicit error bounds and parameter rules, with identified avenues for improving post-processing orders and extending to higher dimensions.

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.

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 work develops a space-time interface-fitted finite element framework for an inverse source problem in an advection-diffusion equation with moving subdomains, regularized by Tikhonov regularization. It proves two second-order -error estimates for the state and adjoint, and analyzes three discretization strategies for the regularized source: variational, element-wise constant, and post-processing, establishing optimal convergence for the first two and a superlinear rate for the post-processing. The paper also provides a priori choices for the regularization parameter (balancing mesh size and data noise ) to guarantee strong convergence of computed sources to the exact one as . The results extend space-time interface techniques to inverse problems with moving interfaces and yield explicit error bounds and parameter rules, with identified avenues for improving post-processing orders and extending to higher dimensions.

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 -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.

Paper Structure

This paper contains 18 sections, 27 theorems, 277 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Problem setting
  3. Function spaces
  4. The advection-diffusion problem
  5. The inverse source problem
  6. Tikhonov regularization
  7. Existence of solutions and optimality conditions
  8. Convergence
  9. Discretization
  10. Space-time interface-fitted method for state and adjoint problems
  11. Discretization of the regularized source
  12. A priori error estimates for advection-diffusion problems
  13. Error estimates and convergence rates for the regularized problem
  14. Variational approach
  15. Element-wise constant discretization
  16. ...and 3 more sections

Key Result

Theorem 3.1

For any fixed $\lambda>0$, Problem eq: problem formulation has a unique solution $f_{\lambda}^{\varepsilon}\in F_{+}$.

Figures (2)

  • Figure 1: The interface $\Gamma(t)$, which envolves by a velocity $\mathbf{v}$, devides the domain $\Omega$ into two subdomains $\Omega_1(t)$ and $\Omega_2(t)$. The subdomain $\omega\subset\Omega_2(t)$ is fixed for all $t\in \left[0,T\right]$, considering $d=2$.
  • Figure 2: The region $S_h=S_h^1\cup S_h^2$ lies between the space-time interface $\Gamma^\ast$ (red) and the discrete one $\Gamma_h^\ast$ (black), considering $d=1$.

Theorems & Definitions (53)

  • Definition 2.1
  • Theorem 3.1
  • proof
  • Theorem 3.2
  • proof
  • Theorem 3.3
  • proof
  • Lemma 3.1
  • Lemma 5.1
  • proof
  • ...and 43 more