with Discontinuous Solution

**By Dr. Daoqi Yang **

** Department of Mathematics**

**Wayne State University**

** Detroit, MI 48202-3483**

`yang@math.wayne.edu`

Many physical problems involve interfaces where two different materials
contact with each other, or singular sources or dipoles are present
along the interfaces immersed in the same fluid.
Such interface problems are modeled by differential equations with
discontinuous coefficients and discontinuous solution.
For example, in multiphase flow of fluids with different
densities and viscosities, and surface tension, the pressure is
discontinuous along the fluid interface due to surface tension, and
the coefficients to the Navier-Stokes equations modeling the fluid
motion are discontinuous across the interface.
The numerical simulation of these problems is a challenging research area since conventional numerical methods fail to capture the discontinuity of the solution, or perform very poorly. They tend to introduce tremendous errors across the interface of discontinuity and either smear out the interface or cause non-physical oscillation in the approximate solution. In this talk, I present a domain decomposition approach to solving these problems. First decompose the physical domain into subdomains separated by the interface and then solve the problems defined on subdomains. Note that the subdomain problems have continuous solution and continuous coefficients (eg, density of the same material) and thus are easy to solve with good accuracy. The proposed method is an iterative process with a given arbitrary initial guess on the interface and iteratively solves the subdomain problems with updated interface conditions to obtain accuracy. The problems defined on subdomains will be solved by finite element or finite difference methods. |

**Forum On Numerical Methods for
Partial Differential Equations**