非定常偏微分方程基于特征投影分解和离散先验插值方法的降阶时空有限元法研究

The non-stationary partial differential equations (NPDEs) have very broad applied background. The project is mainly focused on the optimization numerical methods and their theories based on proper orthogonal decomposition (POD) and discrete empirical interpolation method (DEIM) for NPDEs, with the goal to employ the POD method to simplify the classical space-time finite element methods for NPDEs which include millions or even tens of millions of unknown quantities. Thus, they are simplified into the reduced-order optimized numerical models which only contain a few, up to a dozen unknowns, but have sufficiently high accuracy. We adopt DEIM in algorithm implementations for the reduced-order optimized numerical models such that their computing time is diminished greatly. Thereby, we expect that these methods become a set of new optimized numerical computational theories and methods for NPDEs. And then, we will use the new optimized numerical computational theories and methods for NPDEs to implement numerical simulations for real-life problems. The unknown of the optimized reduced-order numerical models based on POD and DEIM can be greatly diminished. Thus, they can greatly save computational load and computing time as well as memory requirement so that they can greatly reduce the truncation error accumulation in the computational process and improve computational efficiency and computational accuracy. Therefore, the research on the project has not only important theoretical value, but also has wide application prospect.

非定常偏微分方程有着广泛的应用背景。本项目主要针对非定常偏微分方程去做基于特征投影分解(POD)和基于离散先验插值方法的最优化数值计算方法与理论研究。项目的目标是利用POD方法将含有数百万甚至数千万个未知量的非定常偏微分方程经典的时空有限元法，降阶成为只含有很少几个、最多十几个未知量，但具有足够高精确度的最优化降阶数值模型。并在这些最优化降阶数值模型的算法实现中，采用离散先验插值方法使其计算时间极大地减少，使这套最优化数值方法成为非定常偏微分方程的最优化计算新理论和新方法；并用这些非定常偏微分方程的最优化数值计算新理论和新方法对实际问题做数值模拟。这些基于POD和基于离散先验插值方法的最优化降阶数值模型将能极大地减少未知量，从而极大地节省计算量和计算时间及内存的要求，并极大地减少计算过程中截断误差的积累、提高计算效率和计算精确度。因此，该项目既具有重要的理论价值，又具有广泛的的应用前景。

非定常偏微分方程有着广泛的应用背景。本项目已经针对非定常偏微分方程用基于特征投影分解(POD)和基于离散先验插值方法的最优化时空有限元数值计算方法与理论做了大量研究。该项目已利用POD方法将含有数百万甚至数千万个未知量的非定常偏微分方程经典的时空有限元法及相关数值方法降阶成为只含有很少几个、最多十几个未知量但具有足够高精确度的最优化降阶数值模型。并在这些最优化降阶数值模型的算法实现中，采用了离散先验插值方法使其计算时间极大地减少，这套最优化数值方法已经成为非定常偏微分方程的最优化计算新理论和新方法；而且用这些非定常偏微分方程的最优化数值计算新理论和新方法对实际问题做数值模拟。这些基于POD和基于离散先验插值方法的最优化降阶数值模型极大地减少未知量，从而极大地节省计算量和计算时间及内存的要求，并极大地减少计算过程中截断误差的积累、提高计算效率和计算精确度。因此，该项目既具有重要的理论价值，又具有广泛的应用前景。