本项目研究在一致笛卡尔网格上求解界面问题的一种高效而可靠的扩展杂交间断加略金方法。考虑任意一个存在光滑界面的问题，我们将通过在界面附近引入一个间断函数，进而把原问题转换成一个新的界面问题，使得新问题的界面只是出现在网格单元的边界上。然后采用杂交间断伽略金方法来求解这个新的界面问题，以此保证扩展杂交间断加略金方法的高阶收敛性。在数值计算格式中，通过使数值通量在弱的意义下满足适当的跳跃条件，从而精确地保持解在界面上的间断。此外，杂交间断伽略金方法还可以大大减少离散偏微分方程时所产生的全局自由度数目，保证了扩展杂交间断加略金方法的高效性。本项目首先研究相对简单却具有启发意义的椭圆界面问题，而后再利用扩展杂交间断加略金方法的设计思想构造求解复杂流体界面问题数值方法。

This program is to investigate an efficient and robust extended hybridizable discontinuous Galerkin (XHDG) method for solving interface problems on uniform cartesian mesh. Consider an interface problem with arbitrary smooth interface, we introduce an ansatz function to reformulate the original interface problem into a new one in which the inerface conforms to to the element boundary. Then the HDG method is used to solve the new interface problem for hight order convergence rate. In the numerical scheme, jump condition is imposed weakly on the numerical flux to accurately reserve the discontinuity of the solution. Since HDG method can reduce the number of global freedoms, this idea leads to a high efficient numerical method. We first investigate the basic but valuable elliptic interface problems. Then this idea shall be used to design numerical methods for solving complex fluid interface problems.