一种带气泡富集的虚拟区域方法
A Fictitious Domain Formulation with Bubble Enrichment
- Politecnico di Torino（都灵理工大学）
机构由 AI 辅助整理，请以论文原文为准。
AI总结：
本研究提出一种放宽网格尺寸比要求的虚拟区域方法，通过气泡函数富集解空间并构造算子证明均匀离散inf-sup条件，从而保证格式稳定性并给出最优误差估计。
AI中文摘要：
虚拟区域方法通过将复杂物理区域嵌入到更简单的、不匹配的背景网格中，为规避复杂的网格生成过程提供了一种强大的替代方案。在多种在此类网格上施加边界条件的策略中，一种可能性是通过拉格朗日乘子弱施加边界条件。遵循此策略的现有方法通常需要为区域和边界分别使用两个独立的均匀网格，其网格尺寸由预设的比率相关联。在本工作中，我们放宽了这一要求，直接由背景三角剖分的迹构造边界网格。为了在较弱的网格假设下恢复均匀离散inf-sup条件，我们用气泡函数富集解的离散空间。我们的主要结果是构造了一个合适的算子，使我们能够证明该均匀离散inf-sup条件，从而确立所得格式的稳定性。由此，我们推导出最优先验误差估计，并提供数值实验以验证理论结果。
英文摘要：
The fictitious domain method offers a powerful alternative to circumvent complex mesh generation process by embedding the complex physical domain into a simpler, non-matching background mesh. Among the various strategies to enforce the boundary condition on such a mesh, one possibility is to impose it weakly through a Lagrange multiplier. Existing approaches following this strategy typically require two independent uniform meshes for the domain and the boundary, whose mesh sizes are related by a prescribed ratio. In this work, we relax this requirement by constructing the boundary mesh directly from the trace of the background triangulation. To recover the uniform discrete inf-sup condition under this weaker mesh assumption, we enrich the discrete space of the solution with bubble functions. Our main result is the construction of a suitable operator that allows us to prove this uniform discrete inf-sup condition, thereby establishing the stability of the resulting scheme. As a consequence, we derive optimal a priori error estimates and provide a numerical experiment to validate the theoretical results.
