An approximate inertial manifolds approach to postprocessing the Galerkin method for the Navier-Stokes equations
An approximate inertial manifolds approach to postprocessing the Galerkin method for the Navier-Stokes equations
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DOI：
10.1090/s0025-5718-99-01057-1
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发表时间：
1999-07
期刊：
影响因子：
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通讯作者：
B. García-Archilla;J. Novo;E. Titi
中科院分区：
文献类型：
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作者：
B. García-Archilla;J. Novo;E. Titi
在最近的一篇文章中，我们介绍了周期边界条件下耗散发展偏微分方程组的Galerkin方法的后处理过程。后处理技术使用近似惯性流形以Galerkin近似的形式近似精确解中的高阶振型(小尺度分量)，在这种情况下，高阶振型(大尺度分量)起作用。这一过程可以被视为一种缺陷纠正技术。但与标准程序相反的是，只有在时间演化完成时才计算校正。在这里，我们将这些结果推广到更现实的边界条件。具体地说，我们详细研究了齐次(非滑动)Dirichlet边界条件下的二维Navier-Stokes方程。我们还讨论了其他方程，如反应扩散方程和Cahn-Hilliard方程。
In a recent paper we have introduced a postprocessing procedure for the Galerkin method for dissipative evolution partial differential equations with periodic boundary conditions. The postprocessing technique uses approximate inertial manifolds to approximate the high modes (the small scale components) in the exact solutions in terms of the Galerkin approximations, which in this case play the role of the lower modes (large scale components). This procedure can be seen as a defect-correction technique. But contrary to standard procedures, the correction is computed only when the time evolution is completed. Here we extend these results to more realistic boundary conditions. Specifically, we study in detail the two-dimensional Navier-Stokes equations subject to homogeneous (nonslip) Dirichlet boundary conditions. We also discuss other equations, such as reaction-diffusion systems and the Cahn-Hilliard equations.