The Multi-conjugate Adaptive Optics RelaY (MAORY) is one of the key adaptive optics (AO) systems on the European Southern Observatory’s Extremely Large Telescope. MAORY aims to achieve good wavefront correction over a large field of view, which involves a tomographic estimation of the three-dimensional atmospheric wavefront disturbance. Mathematically, the reconstruction of turbulent layers in the atmosphere is severely ill-posed, hence, limits the achievable reconstruction accuracy. Moreover, the reconstruction has to be performed in real time at a few hundred to one thousand hertz frame rates. Huge amounts of data have to be processed and thousands of actuators of the deformable mirrors have to be controlled by elaborated algorithms. Even with extensive parallelization and pipelining, direct solvers, such as the matrix vector multiplication method, are extremely demanding. Thus, research in recent years shifted into the direction of iterative methods. We focus on the iterative finite-element wavelet hybrid algorithm (FEWHA). The key feature of FEWHA is a matrix-free representation of all operators involved, which makes the algorithm fast and enables on-the-fly system updates whenever parameters at the telescope or in the atmosphere change. We provide a performance analysis of the method regarding quality and run-time for the MAORY instrument using the AO software package COMPASS.