半定常可压缩Stokes系统混合方法的收敛性
Convergence of a mixed method for a semi-stationary compressible Stokes system
AI总结：
针对Navier滑移边界条件下的半定常可压缩Stokes系统，提出一种混合有限元方法并证明其收敛到弱解，关键步骤包括速度场的强空间紧致性和DG近似的强收敛性。
摘要：
我们提出并分析了一种用于半定常Stokes系统的有限元方法，该系统模拟了在Navier滑移边界条件下的可压缩流体流动。速度（动量）方程采用最低阶第一类Nedelec空间的混合有限元方法逼近。连续性方程采用标准分片常数迎风间断Galerkin格式逼近。我们的主要结果表明数值方法收敛到弱解。收敛性证明包括两个主要步骤：(i) 建立速度场的强空间紧致性，由于单元空间仅满足div或curl相容性，这一步较为复杂。(ii) 证明间断Galerkin逼近强收敛，考虑到非线性压力函数，这是必需的。分析中涉及的工具包括间断Galerkin逼近的高阶可积性估计、有效粘性通量的离散方程以及间断Galerkin格式的各种重整化形式。
英文摘要：
We propose and analyze a finite element method for a semi-stationary Stokes system modeling compressible fluid flow subject to a Navier-slip boundary condition. The velocity (momentum) equation is approximated by a mixed finite element method using the lowest order Nedelec spaces of the first kind. The continuity equation is approximated by a standard piecewise constant upwind discontinuous Galerkin scheme. Our main result states that the numerical method converges to a weak solution. The convergence proof consists of two main steps: (i) To establish strong spatial compactness of the velocity field, which is intricate since the element spaces are only div or curl conforming. (ii) To prove that the discontinuous Galerkin approximations converge strongly, which is required in view of the nonlinear pressure function. Tools involved in the analysis include a higher integrability estimate for the discontinuous Galerkin approximations, a discrete equation for the effective viscous flux, and various renormalized formulations of the discontinuous Galerkin scheme.
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