广义牛顿流非残差VMS近似的动态子尺度与无条件半离散稳定性
Dynamic subscales and unconditional semi-discrete stability for non-residual VMS approximations of generalised Newtonian flows
- University of Strathclyde（斯特拉斯克莱德大学）
- Universidad de Santiago de Chile（智利圣地亚哥大学）
- Universitat Politècnica de Catalunya（加泰罗尼亚理工大学）
机构由 AI 辅助整理，请以论文原文为准。
AI总结：
该研究针对不可压缩广义牛顿流的非残差变分多尺度有限元格式，通过正交投影分离相关贡献，证明了线性半离散问题的适定性与无条件稳定性，以及非线性格式的存在性与最优阶先验界，明确了动态压力子尺度的关键作用。
AI中文摘要：
我们针对不可压缩广义牛顿Navier--Stokes流，分析了一种带有动态子尺度的非残差变分多尺度有限元格式。表观粘度相对于剪切率是有界且Lipschitz连续的，涵盖多种正则化流变定律。该格式通过正交投影分离了未 resolved 的压力散度、对流项及压力梯度贡献。针对线性化半离散问题，我们在各向异性VMS范数下证明了适定性与无条件稳定性估计，该估计控制了粘性耗散、子尺度能量、离散散度以及耦合的加速度-对流-压力平衡。针对非线性格式，在合适的正则性与小性假设下，通过不动点论证得到了存在性与最优阶先验界。该分析还提供了离散压力-对流平衡的时间弱控制。关键在于，与对应准静态情形不同，动态压力相关子尺度提供了控制投影压力梯度误差所需的时间导数项。
英文摘要：
We analyse a non-residual variational multiscale finite element formulation with dynamic subscales for incompressible generalised Newtonian Navier--Stokes flows. The apparent viscosity is assumed to be bounded and Lipschitz continuous with respect to the shear rate, covering several regularised rheological laws. The method separates, through orthogonal projections, the unresolved pressure-divergence, convective, and pressure-gradient contributions. For the linearised semi-discrete problem, we prove well-posedness and an unconditional stability estimate in an anisotropic VMS norm. The estimate controls viscous dissipation, subscale energies, discrete divergence, and a coupled acceleration-convection-pressure balance. For the non-linear formulation, a fixed-point argument yields existence and optimal-order a priori bounds under suitable regularity and smallness assumptions. The analysis also provides weak-in-time control of the discrete pressure-convection balance. A key point is that the dynamic pressure-related subscale provides the time-derivative contribution needed to control the projected pressure-gradient error, a mechanism unavailable in the corresponding quasi-static setting.
