A Fictitious Domain Formulation with Bubble Enrichment
Abstract
The fictitious domain method offers a powerful alternative to circumvent complex mesh generation process by embedding the complex physical domain into a simpler, non-matching background mesh. Among the various strategies to enforce the boundary condition on such a mesh, one possibility is to impose it weakly through a Lagrange multiplier. Existing approaches following this strategy typically require two independent uniform meshes for the domain and the boundary, whose mesh sizes are related by a prescribed ratio. In this work, we relax this requirement by constructing the boundary mesh directly from the trace of the background triangulation. To recover the uniform discrete inf-sup condition under this weaker mesh assumption, we enrich the discrete space of the solution with bubble functions. Our main result is the construction of a suitable operator that allows us to prove this uniform discrete inf-sup condition, thereby establishing the stability of the resulting scheme. As a consequence, we derive optimal a priori error estimates and provide a numerical experiment to validate the theoretical results.
Related
- A Reduced Coupling-Space Multigrid Schur Preconditioner for Fictitious-Domain Interface Problems
- On the Choice of Continuous and Discontinuous Lagrange Multipliers in Fictitious-Domain Methods
- A unified variational framework for augmented Lagrangian preconditioning of saddle point systems: applications to mixed-dimensional and fictitious domain methods
- A Modified Brinkman-Type Fictitious-Domain Finite Element Method for Incompressible Flow with Rigid Obstacles
- Linear Robin-type Domain Decomposition scheme for Mixed Formulation of Richards' Equation
BibTeX
@misc{berrone2026fictitious,
title = {{A Fictitious Domain Formulation with Bubble Enrichment}},
author = {Stefano Berrone and Lorenzo Neva and Stefano Scialò and Fabio Vicini},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.19836},
eprint = {2609.19836},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.19836},
note = {CC BY-NC-ND 4.0},
}
RIS
TY - UNPB
TI - A Fictitious Domain Formulation with Bubble Enrichment
AU - Stefano Berrone
AU - Lorenzo Neva
AU - Stefano Scialò
AU - Fabio Vicini
PY - 2026
DA - 2026/09/17
PB - arXiv
DO - 10.48550/arxiv.2609.19836
UR - https://arxiv.org/abs/2609.19836
AB - The fictitious domain method offers a powerful alternative to circumvent complex mesh generation process by embedding the complex physical domain into a simpler, non-matching background mesh. Among the various strategies to enforce the boundary condition on such a mesh, one possibility is to impose it weakly through a Lagrange multiplier. Existing approaches following this strategy typically require two independent uniform meshes for the domain and the boundary, whose mesh sizes are related by a prescribed ratio. In this work, we relax this requirement by constructing the boundary mesh directly from the trace of the background triangulation. To recover the uniform discrete inf-sup condition under this weaker mesh assumption, we enrich the discrete space of the solution with bubble functions. Our main result is the construction of a suitable operator that allows us to prove this uniform discrete inf-sup condition, thereby establishing the stability of the resulting scheme. As a consequence, we derive optimal a priori error estimates and provide a numerical experiment to validate the theoretical results.
ER -