A ϕ-FEM approach for time-dependent domains with unfitted meshes
Abstract
In this work, we propose an unfitted finite element scheme to approximate the solution of the heat equation on moving domains. We use the ϕ-FEM paradigm in which the computational domain is described implicitly by a level-set function ϕ. This function is incorporated into the variational formulation in order to enforce the boundary conditions, which allows the use of unfitted meshes in space. Such a strategy avoids the need for remeshing at each time step and makes it possible to handle complex geometrical evolutions. Moreover, the ϕ-FEM approach has the advantage of being simple to implement within standard finite element libraries. We introduce a fully discrete scheme that combines the ϕ-FEM spatial discretization with the lowest-order discontinuous Galerkin method in time. Under regularity assumptions on the level-set function and a mild restriction linking the time step to the mesh size, we establish an optimal a priori error estimate in the L2(0,T;H1) norm. Finally, we present several numerical experiments that confirm this convergence rate, exhibit a second-order convergence in the L{\infty}(0,T;L2) norm, and illustrate the robustness and accuracy of the proposed method.
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BibTeX
@misc{duprez2026ϕfem,
title = {{A ϕ-FEM approach for time-dependent domains with unfitted meshes}},
author = {Michel Duprez and Johan Marguet and Alexei Lozinski and Igor Voulis},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.15630},
eprint = {2609.15630},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.15630},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - A ϕ-FEM approach for time-dependent domains with unfitted meshes
AU - Michel Duprez
AU - Johan Marguet
AU - Alexei Lozinski
AU - Igor Voulis
PY - 2026
DA - 2026/09/14
PB - arXiv
DO - 10.48550/arxiv.2609.15630
UR - https://arxiv.org/abs/2609.15630
AB - In this work, we propose an unfitted finite element scheme to approximate the solution of the heat equation on moving domains. We use the ϕ-FEM paradigm in which the computational domain is described implicitly by a level-set function ϕ. This function is incorporated into the variational formulation in order to enforce the boundary conditions, which allows the use of unfitted meshes in space. Such a strategy avoids the need for remeshing at each time step and makes it possible to handle complex geometrical evolutions. Moreover, the ϕ-FEM approach has the advantage of being simple to implement within standard finite element libraries. We introduce a fully discrete scheme that combines the ϕ-FEM spatial discretization with the lowest-order discontinuous Galerkin method in time. Under regularity assumptions on the level-set function and a mild restriction linking the time step to the mesh size, we establish an optimal a priori error estimate in the L2(0,T;H1) norm. Finally, we present several numerical experiments that confirm this convergence rate, exhibit a second-order convergence in the L{\infty}(0,T;L2) norm, and illustrate the robustness and accuracy of the proposed method.
ER -