Data assimilation for the heat equation using stabilized finite element methods

TL;DR

This paper develops a stabilized finite element method for data assimilation in the heat equation, providing a stable, regularized numerical scheme with error estimates that connect the discrete solution to the continuous model's stability.

Contribution

It introduces a novel stabilized finite element approach for heat equation data assimilation, combining optimization with stability-based regularization techniques.

Findings

Unique solution for the semi-discretized system

Error estimates reflecting finite element approximation and model stability

Application to different data assimilation scenarios with stability analysis

Abstract

We consider data assimilation for the heat equation using a finite element space semi-discretization. The approach is optimization based, but the design of regularization operators and parameters rely on techniques from the theory of stabilized finite elements. The space semi-discretized system is shown to admit a unique solution. Combining sharp estimates of the numerical stability of the discrete scheme and conditional stability estimates of the ill-posed continuous pde-model we then derive error estimates that reflect the approximation order of the finite element space and the stability of the continuous model. Two different data assimilation situations with different stability properties are considered to illustrate the framework. Full detail on how to adapt known stability estimates for the continuous model to work with the numerical analysis framework is given in appendix.