SIAM Journal on Applied Dynamical Systems, Volume 20, Issue 2, Page 853-907, January 2021.
We propose a method to integrate dissipative PDEs rigorously forward in time with the use of the finite element method (FEM). The technique is based on the Galerkin projection on the FEM space and estimates on the residual terms. The proposed approach is illustrated on a periodically forced one-dimensional Burgers equation with Dirichlet conditions. For two particular choices of the forcing we prove the existence of the periodic globally attracting trajectory and give precise bounds on its shape.

中文翻译：

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一维Burgers方程的严格有限元

SIAM应用动力系统杂志，第20卷，第2期，第853-907页，2021年1月。
我们提出了一种使用有限元方法（FEM）严格提前将耗散PDE进行积分的方法。该技术基于FEM空间上的Galerkin投影，并估计残差项。在具有Dirichlet条件的周期强迫一维Burgers方程上说明了所提出的方法。对于强迫的两个特定选择，我们证明了周期性全局吸引轨迹的存在，并给出了其形状的精确界限。