The solution of computational fluid dynamics problems is one of the most computationally hard tasks, especially in the case of complex geometries and turbulent flow regimes. We propose to use Tensor Train (TT) methods, which possess logarithmic complexity in problem size and have great similarities with quantum algorithms in the structure of data representation. We develop the Tensor train Finite Element Method -- TetraFEM -- and the explicit numerical scheme for the solution of the incompressible Navier-Stokes equation via Tensor Trains. We test this approach on the simulation of liquids mixing in a T-shape mixer, which, to our knowledge, was done for the first time using tensor methods in such non-trivial geometries. As expected, we achieve exponential compression in memory of all FEM matrices and demonstrate an exponential speed-up compared to the conventional FEM implementation on dense meshes. In addition, we discuss the possibility of extending this method to a quantum computer to solve more complex problems. This paper is based on work we conducted for Evonik Industries AG.

翻译：计算流体动力学问题的求解是计算难度最大的任务之一，特别是在复杂几何形状和湍流流态的情况下。我们提出使用张量列（TT）方法，该方法在问题规模上具有对数复杂度，并且在数据表示结构上与量子算法高度相似。我们开发了张量列有限元法——TetraFEM——以及用于通过张量列求解不可压缩纳维-斯托克斯方程的显式数值格式。我们在T型混合器的液体混合模拟中测试了该方法，据我们所知，这是首次使用张量方法处理此类非平凡几何形状。正如预期，我们实现了所有有限元矩阵的指数级内存压缩，并在密集网格上展示了相较于传统有限元实现的指数级加速。此外，我们讨论了将该方法扩展到量子计算机以解决更复杂问题的可能性。本文基于我们为赢创工业集团（Evonik Industries AG）开展的工作。