椭圆与Stokes界面问题的Crouzeix-Raviart浸入有限元之间的结构关系
A Structural Relationship Between Crouzeix-Raviart Immersed Finite Elements for Elliptic and Stokes Interface Problems
- School of Mathematics, Southwest Jiaotong University（西南交通大学数学学院）
- Department of Mathematics, Oklahoma State University（俄克拉荷马州立大学数学系）
机构由 AI 辅助整理，请以论文原文为准。
AI总结：
本文建立了椭圆与Stokes界面问题中浸入CR单元间的代数联系，通过行列式分解揭示唯一可解性，并推广至三维，数值实验验证了最优收敛阶。
AI中文摘要：
本文研究了非拟合网格上标量椭圆界面问题和Stokes界面问题的浸入有限元（IFE）空间之间的结构关系。尽管IFE方法已针对单个偏微分方程模型得到广泛发展，但标量与向量IFE空间之间的代数联系尚未被系统探索。我们建立了椭圆界面问题的浸入Crouzeix-Raviart（CR）单元与Stokes界面问题的浸入CR-$P_0$单元在梯度形式和应力形式下的精确唯一可解性关系。通过分析局部IFE矩阵的块结构，我们证明了Stokes IFE矩阵的行列式可分解为相应椭圆IFE矩阵行列式的函数。该分解揭示了两类浸入单元之间内在的代数联系，并通过标量椭圆情形解释了Stokes IFE空间的唯一可解性。同一结果自然推广到三维情形，并显著简化了相应分析。数值实验证实了三维Stokes界面问题中速度和压力的最优收敛阶。
英文摘要：
In this paper, we study the structural relationship between immersed finite element (IFE) spaces for scalar elliptic and Stokes interface problems on unfitted meshes. Although IFE methods have been widely developed for individual partial differential equation models, the algebraic connection between scalar and vector IFE spaces has not been systematically explored. We establish a precise unisolvence relationship between the immersed Crouzeix-Raviart (CR) element for elliptic interface problems and the immersed CR-$P_0$ element for Stokes interface problems in both gradient and stress formulations. By analyzing the block structure of the local IFE matrices, we show that the determinant of the Stokes IFE matrix factorizes in terms of the determinant of the corresponding elliptic IFE matrix. This factorization reveals an intrinsic algebraic link between the two classes of immersed elements and explains the unisolvence of the Stokes IFE spaces through the scalar elliptic case. The same result extends naturally to three dimensions and significantly simplifies the corresponding analysis. Numerical experiments confirm optimal convergence rates for both velocity and pressure in three-dimensional Stokes interface problems.
