# A minimal stabilization procedure for Isogeometric methods on trimmed geometries

## TL;DR

This paper introduces a minimal stabilization technique for Isogeometric Analysis on trimmed geometries, addressing stability issues and ensuring accurate solutions for elliptic PDEs on complex CAD-derived domains.

## Contribution

The paper presents a novel stabilization method for Isogeometric methods on trimmed geometries, improving stability and accuracy in solving elliptic PDEs.

## Key findings

- Stability issues occur with Nitsche's method on trimmed domains.
- The proposed stabilization guarantees well-posedness and optimal error estimates.
- Numerical examples confirm theoretical stability and accuracy improvements.

## Abstract

Trimming is a common operation in CAD, and, in its simplest formulation, consists in removing superfluous parts from a geometric entity described via splines (a spline patch). After trimming the geometric description of the patch remains unchanged, but the underlying mesh is unfitted with the physical object. We discuss the main problems arising when solving elliptic PDEs on a trimmed domain. First we prove that, even when Dirichlet boundary conditions are weakly enforced using Nitsche's method, the resulting method suffers lack of stability. Then, we develop novel stabilization techniques based on a modification of the variational formulation, which allow us to recover well-posedness and guarantee accuracy. Optimal a priori error estimates are proven, and numerical examples confirming the theoretical results are provided.