标题： POD-ROM 方法：连续参数化逼近的误差分析 显示英文标题

标题： POD-ROM methods: error analysis for continuous parametrized approximations

摘要： 本文研究了参数时变偏微分方程（PDEs）的数值逼近问题，采用的是基于正交分解的降阶模型（POD-ROMs）。尽管文献中已有许多关于参数方程降阶模型的研究，但这些方法的完整误差分析仍然是一个挑战。本文提出了一种基于有限差分的新POD方法，并对其进行了分析。我们得到了该新方法的先验界，适用于给定时间区间内任意时刻以及给定参数区间内任意参数值的情况。我们设计的新POD方法使得能够证明点态时间误差估计，而不是通常在POD方法中获得的平均误差界。大多数关于参数方程POD方法的论文仅仅基于不同时间和参数值计算得到的快照，而非它们的差商。我们表明，本文的误差分析也可以涵盖这种情况（我们称之为标准情况）的误差分析。一些数值实验比较了我们的新方法与标准方法，并支持了误差分析的结果。 显示英文摘要

摘要： This paper studies the numerical approximation of parametric time-dependent partial differential equations (PDEs) by proper orthogonal decomposition reduced order models (POD-ROMs). Although many papers in the literature consider reduced order models for parametric equations, a complete error analysis of the methods is still a challenge. We introduce and analyze in this paper a new POD method based on finite differences (respect to time and respect to the parameters that may be considered). We obtain a priori bounds for the new method valid for any value of time in a given time interval and any value of the parameter in a given parameter interval. Our design of the new POD method allow us to prove pointwise-in-time error estimates as opposed to average error bounds obtained typically in POD methods. Most of the papers concerning POD methods for parametric equations are just based on the snapshots computed at different times and parameter values instead of their difference quotients. We show that the error analysis of the present paper can also cover the error analysis of that case (that we call standard). Some numerical experiments compare our new approach with the standard one and support the error analysis.