用于椭圆界面问题的四边形与六面体浸入有限元方法
Quadrilateral and Hexahedral Immersed Finite Element Methods for Elliptic Interface Problems
AI总结：
本文针对椭圆界面问题，提出四边形与六面体网格的节点等参IFE方法，通过合理选择离散通量施加点解决了现有方法的单值性与角度限制问题，建立最优误差估计并经数值实验验证。
AI中文摘要：
浸入有限元（IFE）方法为在非贴合网格上求解界面问题提供了有效框架，其核心思想是通过在特定点施加界面条件来修改标准有限元空间。已知对于节点自由度，线性IFE基函数在三角形单元上的单值性受限于标量扩散系数的非钝角条件；对于张量扩散系数，该限制更为严格，非钝角条件已不足以保证单值性，甚至在矩形网格上，传统双线性IFE基函数也可能无法满足张量扩散系数的单值性要求。本文中，我们开发并分析了一种基于四边形与六面体网格的节点等参IFE方法，证明通过合理选择离散通量条件的施加点可解决单值性问题。关键发现是，与三角形单元不同，四边形与六面体单元的离散通量并非恒定，这为选择施加点以保证单值性提供了灵活性。我们给出了该施加点的系统选择流程，其不仅能确保一般四边形与六面体单元上IFE基函数对标量或张量值扩散系数的单值性，还能保留所得IFE空间的最优逼近性质。所提出的IFE方法具备多项优势：对复杂几何形状的灵活性、无角度限制、适用于张量扩散系数，从而克服了现有矩形与三角形网格节点IFE方法的局限；我们建立了最优误差估计，并通过数值实验验证了该估计。
英文摘要：
Immersed finite element (IFE) methods provide an effective framework for solving interface problems on unfitted meshes. The basic idea underlying IFE methods is to modify standard finite element spaces by enforcing interface conditions at certain points. It is known that, for nodal degrees of freedom, unisolvence of linear IFE basis functions on triangular elements is subject to a non-obtuse-angle condition for scalar diffusion coefficients. This limitation is more severe for tensor diffusion coefficients, for which the non-obtuse-angle condition is no longer sufficient for unisolvence. Even on rectangular meshes, conventional bilinear IFE basis functions may not be unisolvent for tensor diffusion coefficients. In this paper, we develop and analyze a nodal isoparametric IFE method on quadrilateral and hexahedral meshes and show that the unisolvence issue can be overcome by appropriately selecting the enforcement point of the discrete flux condition. The key observation is that, unlike triangular elements, the discrete flux in quadrilateral and hexahedral elements is not constant, which provides the flexibility to select such an enforcement point to ensure unisolvence. We provide a systematic procedure for this selection that not only ensures unisolvence of the IFE basis functions on general quadrilateral and hexahedral elements with either scalar or tensor-valued diffusion coefficients but also preserves the optimal approximation properties of the resulting IFE space. The proposed IFE method offers several advantages: flexibility for complex geometries, the absence of angle restrictions, and applicability to tensor diffusion coefficients, thereby overcoming the limitations of existing nodal IFE methods on rectangular and triangular meshes. Optimal error estimates are established and confirmed by numerical experiments.
