In this work, we illustrate the connection between adaptive mesh refinement for finite element discretized PDEs and the recently developed bi-level regularization algorithm, due to Nguyen (Inverse Problems 40(4): 2024). By adaptive mesh refinement according to data noise, regularization effect and convergence are immediate consequences. We moreover demonstrate its numerical advantages to the classical Landweber algorithm in terms of time and reconstruction quality for the example of the Helmholtz equation in an aeroacoustic setting.