Adaptive Finite Element Method for Interface Problems

该项目的目标是开发和分析具有极端可变或不连续参数的Darcy-Stokes-Brinkman模型的鲁棒自适应有限元方法。提出的研究内容包括：开发一种鲁棒自适应H（div）有限元方法；进行先验误差分析；推导了后验误差估计器的效率界和可靠性界；证明自适应网格细化过程的收敛性；开发一个计算机程序来实现该方法。Darcy-Stokes-Brinkman模型具有广泛的应用，如地表水和地下水相互作用在地球科学，血液循环在健康科学，燃料电池，过滤问题在环境科学。这些问题的解决引起了数学家和工程师的广泛关注，并在Darcy-Stokes-Brinkman模型的数值算法开发和分析方面做了大量工作。这些工作中的大多数将粘度和渗透率视为恒定或接近零的跳跃。然而，对于实际的相关问题，这些物理参数不是不连续的就是高度可变的。物理参数的高度变化所引起的奇异性给数值算法的发展带来了极大的困难。如何获得有效的、鲁棒的数值方法来模拟这些真实的文字问题是一个非常重要的问题。

The objective of the project is to develop and to analyze a robust adaptive finite element method for the Darcy-Stokes-Brinkman model with extremely variable or discontinuous parameters. The proposed research include: developing a robust adaptive H(div) finite element method; conducting a priori error analysis; deriving efficiency and reliability bounds for a posteriori error estimators; proving convergence of adaptive mesh refinement procedures; developing a computer program for implementation of the method.The Darcy-Stokes-Brinkman model has wide range of applications such as surface and sub-surface water interaction in geoscience, blood circulation in health science and, fuel cells, filtration problems in environment science. Solving these problems has attracted a lot of attention from mathematicians and engineers, and many works have been done in developing and analyzing numerical algorithms for Darcy-Stokes-Brinkman model. Majority of these works treat viscosity and permeability as either constant or near zero jump. However, for the practical relevant problems, these physical parameters are either discontinuous or highly variable. Singularity caused by the highly varied physical parameters creates tremendous difficulty in development of numerical algorithms. There is a great interest in obtaining efficient and robust numerical methods to simulate these real word problems.