水平集函数定义区域上的$H^{-1}$最小二乘UnCut有限元法
An $H^{-1}$ least-squares UnCut FEM on domains defined by a level set function
- Department of Applied Mathematics, The Hong Kong Polytechnic University（香港理工大学应用数学系）
机构由 AI 辅助整理，请以论文原文为准。
AI总结：
该研究针对水平集函数定义的曲边区域上的Poisson与Stokes方程，提出一种基于$H^{-1}$残差最小二乘的新型UnCut FEM，无需用户调优惩罚参数即可保证稳定性，严格证明了$H^1$范数下的最优收敛性并通过数值实验验证。
AI中文摘要：
我们针对由水平集函数表示的曲边边界区域上的Poisson方程和Stokes方程，提出了一种新型UnCut有限元法（FEM）。与$\phi$-FEM类似，该方法避免了在边界单元的切割子区域上进行数值积分，但引入了一种新颖的最小二乘公式，可最小化控制方程的$H^{-1}$残差。该公式无需大的稳定参数即可保证稳定性，从而消除了对用户调优惩罚参数的需求，提升了计算的鲁棒性。我们严格证明了UnCut FEM解对于Poisson方程和Stokes方程在$H^1$范数下均具有最优阶收敛性，并通过数值实验验证了理论分析结果。
英文摘要：
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 $ϕ$-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.
