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.

翻译：多功能混合有限元方法最初由Chen和Williams针对等温不可压缩流提出，发表于《Computers & Mathematics with Applications》第80卷（2020年）的“Versatile mixed methods for the incompressible Navier-Stokes equations”。随后，Miller、Chen和Williams将这些方法拓展至非等温不可压缩流，发表于同一期刊第125卷（2022年）的“Versatile mixed methods for non-isothermal incompressible flows”。此类方法的主要优势在于其灵活性。与传统混合方法不同，它们在动量方程和温度方程中保留了散度项。因此，即使在非零散度情况下，格式的优良性质仍得以保持。这使得它们成为拓展至可压缩流的理想选择——在可压缩流中，散度通常不消失。本文最终构建了方法的完全可压缩拓展。此外，我们验证了所得方法在接近不可压缩极限的弱可压缩流以及马赫数接近0.5的强可压缩流中均表现出卓越性能。