In this work, we present a combination of a multigrid approach and the phi-FEM immersed boundary finite element method to reduce its computational cost while preserving its accuracy. To further reduce the numerical cost of the approach, we also propose the combination of the previous technique with some neural network methods. We illustrate the efficiency of these two approaches with numerical test cases in 2D and 3D.
@article{arxiv.2605.13718,
title = {A multigrid and neural network approach to reduce the computational cost of phi-FEM},
author = {Raphaël Bulle and Michel Duprez and Vanessa Lleras and Killian Vuillemot},
journal= {arXiv preprint arXiv:2605.13718},
year = {2026}
}