不可压缩Navier-Stokes方程的能量稳定且动量守恒的混合有限元方法
Energy stable and momentum conserving hybrid finite element method for the incompressible Navier-Stokes equations
AI总结：
提出一种混合方法求解不可压缩Navier-Stokes方程，通过间断压力场和连续拉格朗日乘子实现局部质量守恒、动量守恒及全局能量稳定，并采用等阶插值。
摘要：
提出了一种用于不可压缩Navier-Stokes方程的混合方法。该方法继承了迎风间断伽辽金方法在动量对流显著时的吸引人的稳定机制，速度和压力场可以使用等阶插值，并且质量可以局部守恒。通过使用连续拉格朗日乘子空间来强制单元面上的通量连续性，全局自由度数与该网格上的连续伽辽金方法相同。与我们早期针对Navier-Stokes方程的方法研究不同，本文中的压力场在单元边界上是不连续的。结果表明，这导致了非常好的局部质量守恒，并且对于有限元空间的适当选择，动量也守恒。此外，构建了一种新的动量输运项形式，使得即使在缺乏点式无散速度场的情况下，也能保证全局能量稳定性。对于时间连续情况和完全离散格式，证明了质量守恒、动量守恒和全局能量稳定性。所提出的分析结果得到了一系列数值模拟的支持。
英文摘要：
A hybrid method for the incompressible Navier--Stokes equations is presented. The method inherits the attractive stabilizing mechanism of upwinded discontinuous Galerkin methods when momentum advection becomes significant, equal-order interpolations can be used for the velocity and pressure fields, and mass can be conserved locally. Using continuous Lagrange multiplier spaces to enforce flux continuity across cell facets, the number of global degrees of freedom is the same as for a continuous Galerkin method on the same mesh. Different from our earlier investigations on the approach for the Navier--Stokes equations, the pressure field in this work is discontinuous across cell boundaries. It is shown that this leads to very good local mass conservation and, for an appropriate choice of finite element spaces, momentum conservation. Also, a new form of the momentum transport terms for the method is constructed such that global energy stability is guaranteed, even in the absence of a point-wise solenoidal velocity field. Mass conservation, momentum conservation and global energy stability are proved for the time-continuous case, and for a fully discrete scheme. The presented analysis results are supported by a range of numerical simulations.
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