界面问题浸入有限元方法及其理论分析

项目名称： 界面问题浸入有限元方法及其理论分析

项目编号： No.11471196

项目类型： 面上项目

立项/批准年度： 2015

项目学科： 数理科学和化学

项目作者： 陈焕贞

作者单位： 山东师范大学

项目金额： 70万元

中文摘要： 界面问题刻画了诸如由复杂地质结构或多相流导致的具有间断扩散系数的混溶驱替等实际渗流过程，建立其准确高效的数值模拟方法与完善的数值分析理论体系，对深刻揭示实际渗流的运动机理、指导科学工程实践具有重要的理论价值与应用前景。本项目旨在对具有张量扩散系数（各向异性渗流）的二阶椭圆界面问题，通过强加界面跳跃条件至界面单元空间，构造对界面具有良好辨识能力与逼近性质的Lagrange型与Crouzeix-Raviart型有限元空间，据此，建立沿界面单元边界惩罚的界面浸入有限元方法与相应的最优阶误差估计理论，并推广至具非齐次界面跳跃条件的二阶椭圆界面问题。对具纯量扩散系数（各向同性渗流）的椭圆界面问题有限元误差分析中存在的问题进行改进与修正，弥补当前研究成果的缺失，完善各向同性渗流问题界面浸入有限元方法的收敛性理论体系。进一步，对具有随时间变动界面的二阶抛物型界面问题设计恰当的界面浸入有限元数值模拟格式。

中文关键词： 各向异性渗流；二阶椭圆界面问题；界面浸入有限元方法；收敛性分析；数值模拟

英文摘要： Interface problems describe many practical percolation process such as the miscible displacement with discontinuous diffusion coefficient due to complex strata or multi-phase flux. The establishment of accurate, highly efficient numerical methods and completed numerical analysis has important theoretical value and application aspects for deeply revealing the mechanism of percolation and guiding science engineering practice. The first goal of the project is to simulate numerically the second-order interface elliptic problems with tensor diffusion coefficient(anistropic flow case). By enforcing the jump condition into the finite element space involved the interface element, we construct the piecewise linear finite element space of Lagrange and Crouzeix-Raviart type on each element. Based on the space's capability of recognizing and approximating the interface, we develop the corresponding partially penalty immersed interface finite element methods and their optimal-order error analysis. An extension to second order interface elliptic problems with inhomogeneous jump conditions is made. The second goal is to improve and modify the problems appearing in the previous numerical analysis on the immersed interface finite element methods for the second order interface elliptic problems with scale-function diffusion coefficient (isotropic flow case). By doing so, we complete the optimal-order convergence analysis for the isotropic flow case. Further, we try to design an immersed interface finite element procedures for the second order interface parabolic problems with moving interface.

英文关键词： Anisotropic percolation;Second order elliptic interface problems;Immersed interface finite element method;Covergence Analysis;Numerical simulation