An $H^{-1}$ least-squares UnCut FEM on domains defined by a level set function

Jiashun Hu; Buyang Li; Han Yang

An $H^{-1}$ least-squares UnCut FEM on domains defined by a level set function

Numerical Analysis 2026-08-03 v1

Authors: Jiashun Hu , Buyang Li , Han Yang

Abstract

We propose a novel UnCut finite element method (FEM) for the Poisson and Stokes equations on domains with curved boundaries represented by a level set function. Like the $\phi$ -FEM, the method avoids numerical integration over cut subregions of boundary elements, but introduces a novel least-squares formulation that minimizes an $H^{-1}$ residual of the governing equations. This formulation ensures stability without requiring large stabilization parameters, thereby eliminating the need for user-tuned penalty parameters and improving the robustness of the computation. Optimal-order convergence of the UnCut FEM solutions is rigorously established in the $H^1$ norm for both the Poisson and Stokes equations, and numerical experiments are presented to support the theoretical analysis.

Keywords

finite element method unfitted FEM least-squares formulation level set method Poisson equation Stokes equation

Cite

@article{arxiv.2608.02015,
  title  = {An $H^{-1}$ least-squares UnCut FEM on domains defined by a level set function},
  author = {Jiashun Hu and Buyang Li and Han Yang},
  journal= {arXiv preprint arXiv:2608.02015},
  year   = {2026}
}