摘要: The Navier-Stokes equation coupled with the Darcy equation through interface conditions has attracted scientists’ attention due to its wide range of applications and significant difficulty in the nonlinearity and interface conditions. This presentation discusses a multi-physics domain decomposition method for decoupling the coupled Navier-Stokes-Darcy system with the Beavers-Joseph interface condition. The wellposedness of this system is first showed by using a branch of singular solutions and the existing theoretical results on the Beavers-Joseph interface condition. Then Robin boundary conditions on the interface are constructed based on the physical interface conditions to decouple the Navier-Stokes and Darcy parts of the system. A parallel iterative domain decomposition method is developed according to these Robin boundary conditions and then analyzed for the convergence, especially for the realistic parameters. Numerical examples are presented to illustrate the features of this method and verify the theoretical results.
摘要: When students in computational mathematics start their research work, they usually encounter two critical issues: (1) How to write a good code which is easy to be understood and modified; (2) How to write a good paper which is clear about the novelty, major difficulties, key ideas, and description in details. By using the sketch of a newly designed finite element implementation course at Missouri University of Science and Technology, we will first briefly introduce a general framework and modularization to code for the finite element methods so that the code package is both conceptually clear and easy to be modified for different equations and methods. Then we will use a couple of examples to explain how to highlight the novelty, major difficulties, and key ideas in a paper and how to provide the correct and clear description throughout the paper.