**数学与系统科学研究院**

**计算数学所学术报告**

**报告人**：
Peter D. Minev

University of Alberta, Canada

**报告题目**
Projection Methods for Incompressible Free-Boundary Problems

**Abstract:**
There are several particular issues that make the application of projection methods to free-boundary problems awkward. Probably the most important one is that the very essence of the projection methods is in the separation of the pressure and velocity on one hand, and on the other, they are very essentially coupled via the boundary condition on the capillary free boundaries. The splitting of this condition is nontrivial and it raises problems with the accuracy of the entire scheme as shown by Guermond et al [1]. Improving this splitting is still an open question and we will show some ideas for further work as well as some currently available remedies as suggested in Minev et al [2].

The most widely used approach for solving free-boundary problems at present probably is the so-called Eulerian approach. It raises the other major concern that will be discussed in this presentation. While moving the free boundaries through the stationary grid, it generally intersects some elements. Most of the currently available techniques ignore this and use the usual (locally very smooth) interpolation. This is in contrast with the actual properties of the solution across the free interface and significantly worsens the interpolation error. Some possible remedies of this problem will also be demonstrated. The third issue that we would like to consider is about the solution of the pure advection problem for the motion of the free-boundary that is, in our opinion, the last major challenge in the solution of free-boundary problems. There are several ways for tracking the free boundary including the level-set, the volume-of-fluid and the surface tracking algorithms. But a major problem for each of them when combined with projection methods is that the end-of-step velocity is usually not divergence free and this compromises the solution of the advection problem. A projection method that tries to circumvent this issue will be presented.

In addition, some numerical results illustrating the issues discussed above will be also presented.

**报告时间**：2004年9月10日 下午4：30

**报告地点**：科技综合楼三层报告厅