SIAM Review, Volume 65, Issue 4, Page 1107-1107, November 2023.
The SIGEST article in this issue is “Are Adaptive Galerkin Schemes Dissipative?” by Rodrigo M. Pereira, Natacha Nguyen van yen, Kai Schneider, and Marie Farge. “Although this may seem a paradox, all exact science is dominated by the idea of approximation.” With this quote from Bertrand Russell from 1931 commences this issue's SIGEST article. Indeed, approximation is at the core of mathematics associated to studying partial differential equations (PDEs) with the idea of approximating the solution to the continuous equation with a finite number of modes. The finite element method for PDEs is a prime exemplar of such an approximation, and much research has been dedicated to getting this approximation as accurate and computationally efficient as possible. In this context, adaptive finite element methods and especially Galerkin methods are often the method of choice. Here, typically, when used for solving evolutionary PDEs the number of modes in the Galerkin scheme is fixed over time. In this article, the authors consider adaptive Galerkin schemes in which the number of modes can change over time, and they introduce a mathematical framework for studying evolutionary PDEs discretized with these schemes. In particular, they show that the associated projection operators, i.e., the operators that project the continuous solution onto the finite-dimensional finite element spaces, are discontinuous and introduce energy dissipation. That this is a significant result is demonstrated by studying adaptive Galerkin schemes for the time evolution of the inviscid Burgers equation in 1D and the incompressible Euler equations in 2D and 3D. They show that adaptive wavelet schemes regularize the solution of the Galerkin truncated equations and yield convergence towards the exact dissipative solution for the inviscid Burgers equation. Also for the Euler equations this regularizing effect can be numerically observed though no exact reference solutions are available in this case. This motivates, in particular, adaptive wavelet Galerkin schemes for nonlinear hyperbolic conservation laws and leave their systematic study for this class of PDEs for an interesting future work. For the SIGEST article the authors have expanded their original Multiscale Modeling & Simulation article by providing a more comprehensive discussion on adaptive Galerkin methods fit for a general mathematical audience. They have also added a new section on continuous wavelet analysis of the inviscid Burgers equation, analyzing its time evolution, and added an illustration for the development of thermal resonances in wavelet space. Overall, adaptive Galerkin methods and their mathematical properties will be of interest to a wide range of applied mathematicians who study PDE models, and also to applied analysts and numerical analysts who wish to simulate PDEs numerically.

中文翻译：

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西格斯特

SIAM Review，第 65 卷，第 4 期，第 1107-1107 页，2023 年 11 月。
本期最重要的文章是“自适应伽辽金方案是耗散的吗？” 作者：Rodrigo M. Pereira、Natacha Nguyen van Yan、Kai Schneider 和 Marie Farge。“虽然这看起来似乎是一个悖论，但所有精确科学都受近似概念的支配。” 本期最重要的文章以 1931 年伯特兰·罗素 (Bertrand Russell) 的这句话开始。事实上，近似是与研究偏微分方程 (PDE) 相关的数学核心，其思想是用有限数量的模态来近似连续方程的解。偏微分方程的有限元方法是这种近似的主要范例，并且许多研究致力于使这种近似尽可能准确和计算高效。在这种情况下，自适应有限元方法，尤其是伽辽金方法通常是首选方法。这里，通常，当用于求解演化偏微分方程时，伽辽金方案中的模式数量随时间的推移是固定的。在本文中，作者考虑了模态数量可以随时间变化的自适应伽辽金方案，并引入了一个数学框架来研究用这些方案离散化的演化偏微分方程。特别是，他们表明相关的投影算子，即将连续解投影到有限维有限元空间上的算子，是不连续的并引入能量耗散。通过研究一维无粘性 Burgers 方程和 2D 和 3D 不可压缩欧拉方程的时间演化的自适应伽辽金格式，证明了这是一个重要的结果。他们表明，自适应小波格式正则化了伽辽金截断方程的解，并收敛于无粘性伯格斯方程的精确耗散解。同样对于欧拉方程，这种正则化效应可以在数值上观察到，尽管在这种情况下没有精确的参考解。这特别激发了非线性双曲守恒定律的自适应小波伽辽金方案，并为此类偏微分方程的系统研究留下了有趣的未来工作。在 SIGEST 文章中，作者对适合一般数学读者的自适应伽辽金方法进行了更全面的讨论，扩展了他们原来的多尺度建模与仿真文章。他们还添加了关于无粘性 Burgers 方程的连续小波分析的新部分，分析其时间演化，并添加了小波空间中热共振发展的说明。总体而言，自适应伽辽金方法及其数学特性将引起研究偏微分方程模型的广泛应用数学家以及希望对偏微分方程进行数值模拟的应用分析师和数值分析师的兴趣。