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.

中文翻译：

---

修剪几何体上等几何方法的最小稳定程序

修剪是 CAD 中的常见操作，在其最简单的公式中，包括从通过样条（样条补丁）描述的几何实体中去除多余的部分。修剪后，面片的几何描述保持不变，但底层网格不适合物理对象。我们讨论在修剪域上求解椭圆偏微分方程时出现的主要问题。首先，我们证明，即使使用 Nitsche 方法弱执行 Dirichlet 边界条件，所得方法也缺乏稳定性。然后，我们基于变分公式的修改开发了新的稳定技术，这使我们能够恢复适定性并保证准确性。证明了最佳先验误差估计，并提供了证实理论结果的数值例子。