A multigrid and neural network approach to reduce the computational cost of phi-FEM

TL;DR

This paper combines multigrid, phi-FEM, and neural networks to reduce computational costs of immersed boundary finite element methods while maintaining accuracy, demonstrated through 2D and 3D tests.

Contribution

It introduces a novel integration of multigrid, phi-FEM, and neural networks to improve efficiency in immersed boundary finite element computations.

Findings

Multigrid and phi-FEM combination reduces computational cost.

Neural network integration further enhances efficiency.

Numerical tests confirm accuracy preservation in 2D and 3D.

Abstract

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.