# CutFEM without cutting the mesh cells: a new way to impose Dirichlet and Neumann boundary conditions on unfitted meshes

## TL;DR

This paper introduces a novel CutFEM approach that avoids integrating over cut elements, maintaining optimal convergence rates for Poisson problems with Dirichlet or Neumann boundary conditions on unfitted meshes.

## Contribution

It proposes a new method that eliminates the need for integration over cut elements in CutFEM, simplifying implementation while preserving accuracy.

## Key findings

- Achieves optimal convergence rates theoretically.
- Numerical results confirm the method's effectiveness.
- Simplifies CutFEM implementation without loss of accuracy.

## Abstract

We present a method of CutFEM type for the Poisson problem with either Dirichlet or Neumann boundary conditions. The computational mesh is obtained from a background (typically uniform Cartesian) mesh by retaining only the elements intersecting the domain where the problem is posed. The resulting mesh does not thus fit the boundary of the problem domain. Several finite element methods (XFEM, CutFEM) adapted to such meshes have been recently proposed. The originality of the present article consists in avoiding integration over the elements cut by the boundary of the problem domain, while preserving the optimal convergence rates, as confirmed by both the theoretical estimates and the numerical results.