非符合结构保持有限元法用于有界Lipschitz域上的双扩散流
Non-Conforming Structure Preserving Finite Element Method for Doubly Diffusive Flows on Bounded Lipschitz Domains
AI总结：
本文研究了在二维和三维有界Lipschitz域上温度依赖粘度的双扩散流稳态模型，提出了一种基于Crouzeix-Raviart元的完全非符合有限元法，确保局部精确无散速度场，并证明了离散解的唯一性及误差估计。
摘要：
我们研究了在二维和三维有界Lipschitz域上温度依赖粘度的双扩散流稳态模型。在最小的域几何和数据正则性假设下，建立了弱解的适定性和正则性分析。基于Crouzeix-Raviart元的完全非符合有限元法被探索，该方法确保局部精确无散速度场。与之前提出的方案不同，这种离散化方法能够证明离散解的唯一性。我们证明了离散问题的适定性，并推导了先验误差估计。进行了精度测试以验证在流体、斯托克斯和达西制度下在凸和非凸域中的理论误差衰减率，并进行了 porous cavity 流的基准测试，将所提方法与现有文献进行比较。
英文摘要：
We study a stationary model of doubly diffusive flows with temperature-dependent viscosity on bounded Lipschitz domains in two and three dimensions. A new well-posedness and regularity analysis of weak solutions under minimal assumptions on domain geometry and data regularity are established. A fully non-conforming finite element method based on Crouzeix-Raviart elements, which ensures locally exactly divergence-free velocity fields is explored. Unlike previously proposed schemes, this discretization enables to establish uniqueness of the discrete solutions. We prove the well-posedness of the discrete problem and derive a priori error estimates. An accuracy test is conducted to verify the theoretical error decay rates in flow, Stokes and Darcy regimes on convex and non-convex domains, and a benchmark test of flow in a porous cavity is conducted, comparing the proposed method with existing literature.
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