arXiv subjects

Killian Vuillemot

Publications and source records attributed to Killian Vuillemot.

6 recordsLinked to original sources

arXiv2026

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

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.

math.NA↗

arXiv2026

A penalized ϕ-FEM scheme for the Poisson Dirichlet problem

In this work, we analyze a penalized variant of the ϕ-FEM scheme for the Poisson equation with Dirichlet boundary conditions. The ϕ-FEM is a recently introduced unfitted finite element method based on a level-set description of the geometry, which avoids the need for boundary-fitted meshes. Unlike the original ϕ-FEM formulation, the method proposed here enforces boundary conditions through a penalization term. This approach has the advantage that the level-set function is required only on the cells adjacent to the boundary in the variational formulation. The scheme is stabilized using a ghost penalty technique. We derive a priori error estimates, showing optimal convergence in the H1 semi-norm and quasi-optimal convergence in the L2 norm under suitable regularity assumptions. Numerical experiments are presented to validate the theoretical results and to compare the proposed method with both the original ϕ-FEM and the standard fitted finite element method.

math.NA↗

arXiv2025

φ-FD : A well-conditioned finite difference method inspired by φ-FEM for general geometries on elliptic PDEs

This paper presents a new finite difference method, called φ-FD, inspired by the ϕ-FEM approach for solving elliptic partial differential equations (PDEs) on general geometries. The proposed method uses Cartesian grids, ensuring simplicity in implementation. Moreover, contrary to the previous finite difference scheme on non-rectangular domain, the associated matrix is well-conditioned. The use of a level-set function for the geometry description makes this approach relatively flexible. We prove the quasi-optimal convergence rates in several norms and the fact that the matrix is well-conditioned. Additionally, the paper explores the use of multigrid techniques to further accelerate the computation. Finally, numerical experiments in both 2D and 3D validate the performance of the φ-FD method compared to standard finite element methods and the Shortley-Weller approach.

math.NA↗

arXiv2025

Phi-FEM-FNO: a new approach to train a Neural Operator as a fast PDE solver for variable geometries

In this paper, we propose a way to solve partial differential equations (PDEs) by combining machine learning techniques and the finite element method called Phi-FEM. For that, we use the Fourier Neural Operator (FNO), a learning mapping operator. The purpose of this paper is to provide numerical evidence to show the effectiveness of this technique. We will focus here on the resolution of two equations: the Poisson-Dirichlet equation and the non-linear elasticity equations. The key idea of our method is to address the challenging scenario of varying domains, where each problem is solved on a different geometry. The considered domains are defined by level-set functions due to the use of the Phi-FEM approach. We will first recall the idea of $φ$-FEM and of the Fourier Neural Operator. Then, we will explain how to combine these two methods. We will finally illustrate the efficiency of this combination with some numerical results on three test cases. In addition, in the last test case, we propose a new numerical scheme for hyperelastic materials following the Phi-FEM paradigm.

math.NA↗

arXiv2023

phi-FEM for the heat equation: optimal convergence on unfitted meshes in space

Thanks to a finite element method, we solve numerically parabolic partial differential equations on complex domains by avoiding the mesh generation, using a regular background mesh, not fitting the domain and its real boundary exactly. Our technique follows the phi-FEM paradigm, which supposes that the domain is given by a level-set function. In this paper, we prove a priori error estimates in l2(H1) and linf(L2) norms for an implicit Euler discretization in time. We give numerical illustrations to highlight the performances of phi-FEM, which combines optimal convergence accuracy, easy implementation process and fastness.

math.NA↗

arXiv2021

$ϕ$-FEM: an efficient simulation tool using simple meshes for problems in structure mechanics and heat transfer

One of the major issues in the computational mechanics is to take into account the geometrical complexity. To overcome this difficulty and to avoid the expensive mesh generation, geometrically unfitted methods, i.e. the numerical methods using the simple computational meshes that do not fit the boundary of the domain, and/or the internal interfaces, have been widely developed. In the present work, we investigate the performances of an unfitted method called $ϕ$-FEM that converges optimally and uses classical finite element spaces so that it can be easily implemented using general FEM libraries. The main idea is to take into account the geometry thanks to a level set function describing the boundary or the interface. Up to now, the $ϕ$-FEM approach has been proposed, tested and substantiated mathematically only in some simplest settings: Poisson equation with Dirichlet/Neumann/Robin boundary conditions. Our goal here is to demonstrate its applicability to some more sophisticated governing equations arising in the computational mechanics. We consider the linear elasticity equations accompanied by either pure Dirichlet boundary conditions or by the mixed ones (Dirichlet and Neumann boundary conditions co-existing on parts of the boundary), an interface problem (linear elasticity with material coefficients abruptly changing over an internal interface), a model of elastic structures with cracks, and finally the heat equation. In all these settings, we derive an appropriate variant of $ϕ$-FEM and then illustrate it by numerical tests on manufactured solutions. We also compare the accuracy and efficiency of $ϕ$-FEM with those of the standard fitted FEM on the meshes of similar size, revealing the substantial gains that can be achieved by $ϕ$-FEM in both the accuracy and the computational time.

math.NA↗