AI 中文总结
本研究提出广义高阶极小化基多项式修正方法，推广SBM与ROD技术，可提升非匹配谱元法的系统条件数，适用于Neumann和Robin边界，经泊松问题数值实验验证其高阶精度。
AI 中文摘要
高阶有限元方法是求解偏微分方程的有效手段，但将其应用于复杂曲域时，常因生成高质量曲边网格的挑战而难以实施。非匹配（或称嵌入）方法通过避免完整网格生成及曲边单元积分，提供了一种可行替代方案，不过其精度会受嵌入过程引入的几何误差影响。本研究探索一类新的多项式修正族，该修正通过求解局部极小化问题得到，旨在提升嵌入边界方法的一致性。该方法推广了现有技术，如偏移边界方法（SBM）和离位数据重构（ROD）方法。与依赖截断泰勒展开的SBM不同，所提出的多项式修正族与ROD方法类似，源自约束极小化问题。研究表明，该方法相比原始SBM能获得更优的系统条件数，且提供了灵活框架，可逐点应用，无需为每个边界单元求解完整的ROD线性系统。本文提出了基于不同泛函的四种形式，并将该极小化基多项式修正族扩展至处理 Neumann 和 Robin 边界条件，证明在此场景下可得到简洁的形式。针对泊松问题开展了若干数值实验，结果显示广义多项式修正在所有边界条件下均能实现高阶精度。
英文摘要
Higher-order finite element methods are effective for solving partial differential equations. However, applying them in complex curved domains is often difficult due to challenges in creating high-quality curvilinear meshes. Unfitted, or embedded, methods provide a valid alternative by avoiding complete mesh generation and curved element integration, but their accuracy can suffer from geometric errors introduced during embedding. In this work, we explore a new family of polynomial corrections obtained by solving a local minimization problem to improve the consistency of embedded boundary methods. This approach generalizes existing techniques such as the shifted boundary method (SBM) and the reconstruction for off-site data (ROD) method. Unlike the SBM, which depends on truncated Taylor expansions, the proposed family of polynomial corrections is derived from a constrained minimization problem, similar to the ROD method. We demonstrate that this approach yields better system conditioning than the original SBM and provides a flexible framework that can be applied pointwise, without solving the full ROD linear system for each boundary element. This paper presents four formulations based on different functionals and extends the family of minimization-based polynomial corrections to handle Neumann and Robin boundary conditions, demonstrating that an elegant formulation is possible in this context. Several numerical experiments for the Poisson problem are presented to show that the generalized polynomial corrections achieve high-order accuracy across all boundary conditions.
Comments11 figures, 28 pages, 56 references
