Phi-FEM: an optimally convergent and easily implementable immersed boundary method for particulate flows and Stokes equations

TL;DR

Phi-FEM is a finite element immersed boundary method that accurately simulates rigid particles in Stokes flows using level-set descriptions, offering optimal convergence and ease of implementation.

Contribution

The paper introduces Phi-FEM, a novel immersed boundary method that combines standard finite element spaces with level-set functions for efficient and accurate particulate flow simulations.

Findings

Achieves optimal convergence in H1 norm for velocity and L2 norm for pressure.

Uses standard finite element spaces and classical integration tools.

Demonstrates numerical convergence consistent with theoretical predictions.

Abstract

We present an immersed boundary method to simulate the creeping motion of a rigid particle in a fluid described by the Stokes equations discretized thanks to a finite element strategy on unfitted meshes, called Phi-FEM, that uses the description of the solid with a level-set function. One of the advantages of our method is the use of standard finite element spaces and classical integration tools, while maintaining the optimal convergence (theoretically in the H1 norm for the velocity and L2 for pressure; numerically also in the L2 norm for the velocity).