求解二阶椭圆界面问题的浸入界面有限元及其多水平方法

In the scientific and engineering computing, we often need to solve the interface problems, such as reservoir simulation, the simulation of epitaxial thin ?lm growth, composite materials, the simulation of ion propulsion and biomechanical problems. It is difficult to find high performance numerical solutions for these problems, due to the complexity of the interface, the high discontinuity of the coefficient, the discontinuity of the solution or its gradient and so on. Therefore, it is quite meaningful to find some high performance discrete methods, and design easy and fast algebraic algorithms, of which the convergence rate is independent of or weakly depends on the jump of the coefficient, the mesh size and the geometry of the interface, for the corresponding linear systems. The main subject of this research is to design and analyze the immersed interface finite element method and the corresponding multilevel methods (including algebraic multilevel methods) and multigrid methods for the second order elliptic interface problems.

在科学与工程计算中，人们经常需要求解界面问题，例如油藏模拟、晶体增长、离子推进器的设计以及生物力学问题等等，其界面几何形状的复杂性、系数的高间断性、解或解的梯度的间断性等特点给该问题数值求解带来了很大的困难。因此研究界面问题的高效的离散方法并且设计相应的简单易算的快速数值求解方法, 即收敛率与间断系数、界面形状以及网格尺寸无关或依赖很弱的算法, 是一个非常有意义的研究课题。本课题主要研究二阶椭圆界面问题的浸入界面有限元方法及其多水平(包括代数多水平)和多重网格算法。

界面问题在很多方面都有很重要的应用，例如油藏模拟、生物力学等等. 基于非界面匹配网格，我们研究了界面问题的混合有限元方法，在一定条件下证明了该方法的最优误差估计与系数跳跃和界面无关. 我们提出了基于非匹配网格的自适应mortar有限元离散系统的局部多水平预处理方法，并证明了该方法与间断系数无关，仅依赖于交叉点附近的网格层数.