具有间断粘性系数的不可压缩流动问题的自适应有限元方法

Solving the incompressible flows with discontinuous viscosity is a fundamental problem of the simulation of multiphase flows. Both the discontinuous viscosity and the incompressibility of flows make such problem hard to solve. ..We will carry out the project as follows. Firstly, propose a general stability for the incompressible flows with discontinuous viscosity on multi-domains. Then, use widely-applied low order finite elements spaces to discrete the problem and provide the stabilized finite element methods and their stable analysis. Secondly, consider the weighted norm and special recovery spaces to design reliable and efficient residual and recovery based error estimators for the non-smooth interface problem. Then, propose the a posteriori error estimate and numerical examples to verify their reliability and efficiency. Finally, combine the iterative algorithm for weighted saddle problems and h-refinement adaptive algorithm to propose an adaptive finite element method... The interest of this project is to propose an efficient and reliable adaptive finite element method for incompressible flows with discontinuous viscosity. Our work would supplement the theories and methods for the domain of computational fluid dynamics. Meanwhile, it would provide the research on multiphase flows with new thoughts and technological assistances.

具有间断粘性系数的不可压缩流动问题来源于多相流的模拟。此问题的研究有助于揭示多相流动特点并预测流动过程。粘性系数的间断性和流体的不可压缩性增加了其求解难度。. 本项目将开展如下研究：首先，研究粘性系数在多个子区域上不同的不可压缩流动问题，提出更具普遍适用性的稳定性结论。采用应用广泛的低阶有限元空间将问题离散，给出稳定化有限元方法及其稳定性分析。其次，采用粘性系数加权的范数和特殊的恢复空间，构造残量型和恢复型误差估计子，并通过后验误差估计和数值算例验证有效性和可靠性。最后，优化结合加权鞍点问题的迭代算法与有限元网格自动加密策略，设计并实现高效自适应有限元方法。. 本项目通过深入探讨间断粘性系数不可压缩流体问题的稳定性分析方法和后验误差估计技巧，提供有效灵活求解问题的自适应有限元方法。研究成果将进一步丰富计算流体力学领域的理论和方法，同时为多相流研究提供新的思路与技术支持。

本项目研究了定义在多个子区域上的具有间断系数的静态Stokes方程的稳定性问题，发现此问题的稳定性是依赖于倒数第二小的粘性系数，并做了严格的理论证明。采用应用广泛的低阶有限元空间将问题离散，给出粘性加权的稳定化有限元方法及其稳定性分析。针对流体流动方程Navier-Stokes问题，研究基于恢复型误差估计子的自适应算法，给出NS方程的非线性项的合理估计并验证了方法的有效性和可靠性，计算结果理想。针对流体界面问题研究了其扩展有限元方法，调整形状函数给出改进的内扩展和外扩展有限元方法，并基于单位分解给出局部并行算法。本项目研究成果有助于揭示多相流流动过程，为多相流研究提供新的思路和技术支持。