Reformulated weak formulation and efficient fully-discrete finite element method for a two-phase ferrohydrodynamics Shliomis model 

报告题目Reformulated weak formulation and efficient fully-discrete finite element method for a two-phase ferrohydrodynamics Shliomis model 


报告时间:2023年5月19日 10:30—11:30



The two-phase ferrohydrodynamics model consisting of the Cahn-Hilliard equations, Navier-Stokes equations,  magnetization equation, and magnetostatic equation is a highly nonlinear, coupled, and saddle point structural multi-physics PDE system.  While various works exist to develop fully decoupled, linear, second-order in time, and unconditionally energy stable methods for simpler gradient flow models, existing ideas may not be applicable to this complex model or may be only applicable to part of this model. Therefore, significant challenges remain in developing {\color{blue}the}  corresponding efficient fully-discrete numerical algorithms with the four above-mentioned desired properties, which will be addressed in this paper by dynamically incorporating several key ideas, including a reformulated weak formulation with special test functions for overcoming two major difficulties caused by the magnetostatic equation, the decoupling technique based on the ``zero-energy-contribution" property to handle the coupled nonlinear terms, the second-order projection method for the Navier-Stokes equations, and the Invariant Energy Quadratization (IEQ) method for the time marching. Among all these ideas, the reformulated weak formulation serves as a key bridge between the existing techniques and the challenges of the target model, with all the four desired properties kept in mind. We demonstrate the well-posedness of the proposed scheme and rigorously show that the scheme is unconditionally energy stable. Extensive numerical simulations, including accuracy/stability tests, and several 2D/3D benchmark Rosensweig instability problems for ``spiking" phenomena of ferrofluids are performed to verify the effectiveness of the scheme.

专家简介:何晓明, 2002年与2005年在四川大学数学学院分别获学士与硕士学位, 2009年在弗吉尼亚理工大学数学系获博士学位,2009年至2010年在佛罗里达州立大学作博士后。2010年至2016年在美国密苏里科技大学任助理教授,2016年晋升为副教授并获终身教职,2021年晋升为正教授。2018年获得Humboldt Research   Fellowship for Experienced Researchers。担任计算数学领域国际期刊International Journal of   Numerical Analysis & Modeling的Managing editor。从2012年起主持了多项由美国国家科学基金会和美国能源部资助的科研项目。2014-2016年担任SIAM美国中部分会的第一任主席和前两届年会的组织委员会主席。2019年起担任Midwest Numerical Analysis Day的组织委员成员。2021年1月起担任SIAM Committee on Programs and Conferences成员。2021年7月起担任密苏里科技大学Faculty Fellow to Vice Chancellor of Research   and Innovation。何晓明教授主要的研究领域是计算科学与工程。研究问题主要包括界面问题,计算流体力学,计算电磁学,有限元方法,各类解耦算法,数据同化,随机偏微分方程,控制问题等。他将计算数学与实际工程应用问题结合起来,在科学计算和应用领域做了大量的工作,在SIAM Journal on Scientific Computing,Journal of Computational Physics,Computer Methods in Applied Mechanics and Engineering,   SIAM Journal on Numerical Analysis, Mathematics of   Computation,Numerische Mathematik,IEEE Transactions on Plasma Science, Lab