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.
