A Divergence Free H(div) Finite Element Method

该项目的目标是发展一种求解Navier-Stokes方程的无散度H(Div)有限元方法，并为该方法的三维H(Div)元提供无散度的子空间基。PI和其他研究人员已经研究了不可压缩流动问题的H(Div)协调元有限元格式。H(Div)有限元方法具有精确满足不可压缩约束的优点。此外，如果H(Div)元素的无散子空间的基是可用的，则速度可以在无散子空间中近似。这消除了来自大型复杂系统的压力和整个不可压缩性约束。因此，构造H(Div)元的精确散度自由子空间基具有重要的理论和实际意义。本文的研究将为一大类具有特殊性质的问题提供一种高效、准确的算法：精确地满足无散度条件。偏微分方程组可能存在于科学和工程的所有领域。研究偏微分方程解的数值解在实际中具有十分重要的意义。一大类来自不同领域的偏微分方程具某些有趣的变量是无散度的约束。例如，这类方程出现在不可压缩流动问题的研究中，已经发展了许多数值方法来解决这类问题。然而，它们中很少满足精确的无散度条件。所提出的研究将为工程技术人员提供一种有前途的数值算法，解决计算流体力学和其他领域中出现的无散度约束问题。建议的项目注重促进教学、培训和学习。

The objectives of the project are to develop a divergence free H(div) finite element method for the Navier-Stokes equations and provide divergence free subspace bases for the three dimensional H(div) elements for the method. The finite element formulations using H(div) conforming elements for the incompressible flow problems have been studied by the PI and other researchers. The H(div) finite element methods have the advantage that the incompressibility constraint is satisfied exactly. Furthermore, if a basis of the divergence free subspace for the H(div) element is available, the velocity can be approximated in the divergence free subspace. This eliminates the pressure and the whole incompressibility constraint from the large complicated system. As a result, the computational cost is reduced significantly.Therefore construction of the exactly divergence free subspaces bases for the H(div) elements has theoretical and practical significance. The proposed research will lead to an efficient and accurate algorithm for a large class of the problems with a special feature: the divergence free condition is satisfied exactly.Partial differential equations may arise in all fields of science and engineering. The study of numerical solutions for solving partial differential equations is extremely important in practice. A large class of the partial differential equations derived from different fields has the constraint that some interesting variable is divergence free. For example, such equations arise in the study of incompressible flow problems.Many numerical methods have been developed for solving this kind of problem. However, very few of them satisfy the exact divergence free condition. The proposed research will provide engineers with a promising numerical algorithm for the problems arising from computational fluid dynamics and other fields where the divergence free constraint arises. The proposed project pays intensive attention to promoting teaching, training, and learning.