Versatile mixed finite element methods were originally developed by Chen and Williams for isothermal incompressible flows in "Versatile mixed methods for the incompressible Navier-Stokes equations," Computers & Mathematics with Applications, Volume 80, 2020. Thereafter, these methods were extended by Miller, Chen, and Williams to non-isothermal incompressible flows in "Versatile mixed methods for non-isothermal incompressible flows," Computers & Mathematics with Applications, Volume 125, 2022. The main advantage of these methods lies in their flexibility. Unlike traditional mixed methods, they retain the divergence terms in the momentum and temperature equations. As a result, the favorable properties of the schemes are maintained even in the presence of non-zero divergence. This makes them an ideal candidate for an extension to compressible flows, in which the divergence does not generally vanish. In the present article, we finally construct the fully-compressible extension of the methods. In addition, we demonstrate the excellent performance of the resulting methods for weakly-compressible flows that arise near the incompressible limit, as well as more strongly-compressible flows that arise near Mach 0.5.

中文翻译：

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可压缩流的多功能混合方法

多功能混合有限元方法最初由 Chen 和 Williams 在“不可压缩纳维-斯托克斯方程的多功能混合方法”中针对等温不可压缩流动开发，计算机与数学与应用，第 80 卷，2020 年。此后，这些方法由 Miller 扩展， Chen 和 Williams 在“非等温不可压缩流的通用混合方法”中对非等温不可压缩流进行了研究，计算机与数学与应用，第 125 卷，2022 年。这些方法的主要优点在于其灵活性。与传统的混合方法不同，它们保留了动量和温度方程中的发散项。因此，即使存在非零散度，该方案的有利特性也得以保持。这使它们成为可压缩流扩展的理想候选者，其中发散通常不会消失。在本文中，我们最终构建了该方法的完全可压缩扩展。此外，我们还展示了所得方法对于不可压缩极限附近出现的弱可压缩流以及在 0.5 马赫附近出现的更强可压缩流的优异性能。