Gopalakrishnan J, Kogler L, Lederer P L, Schöberl J
Portland State University, PO Box 751, Portland, OR 97207 USA.
Institute for Analysis and Scientific Computing, TU Wien, Wiedner Hauptstraße 8-10, 1040 Wien, Austria.
J Sci Comput. 2023;95(3):91. doi: 10.1007/s10915-023-02203-8. Epub 2023 May 11.
We introduce two new lowest order methods, a mixed method, and a hybrid discontinuous Galerkin method, for the approximation of incompressible flows. Both methods use divergence-conforming linear Brezzi-Douglas-Marini space for approximating the velocity and the lowest order Raviart-Thomas space for approximating the vorticity. Our methods are based on the physically correct viscous stress tensor of the fluid, involving the symmetric gradient of velocity (rather than the gradient), provide exactly divergence-free discrete velocity solutions, and optimal error estimates that are also pressure robust. We explain how the methods are constructed using the minimal number of coupling degrees of freedom per facet. The stability analysis of both methods are based on a Korn-like inequality for vector finite elements with continuous normal component. Numerical examples illustrate the theoretical findings and offer comparisons of condition numbers between the two new methods.
我们引入了两种新的最低阶方法，一种混合方法和一种混合间断伽辽金方法，用于不可压缩流的近似。两种方法都使用散度协调的线性布雷齐 - 道格拉斯 - 马里尼空间来近似速度，并用最低阶拉维尔特 - 托马斯空间来近似涡度。我们的方法基于流体物理上正确的粘性应力张量，涉及速度的对称梯度（而非梯度），能提供精确的无散度离散速度解以及压力鲁棒的最优误差估计。我们解释了如何使用每个面最少数量的耦合自由度来构建这些方法。两种方法的稳定性分析都基于具有连续法向分量的向量有限元的类科恩不等式。数值例子说明了理论结果，并对两种新方法的条件数进行了比较。
