In this talk, we will present a hybrid mixed finite element method for poroelasticity model. The hybrid P1-RT0-P0 discretization of the displacement-pressure-Darcy’s velocity system of Biot’s model presented is not uniformly stable with respect to the physical parameters, resulting in some issues in numerical simulations. To alleviate such problems,  we stabilize the hybrid scheme with face bubble functions and show the well-posedness with respect to physical and discretization parameters, which provide optimal error estimates of the stabilized method. We introduce a perturbation of the bilinear form of the displacement which allows for the elimination of the bubble functions. Together with eliminating Darcy’s velocity by hybridization, we obtain an eliminated system whose size is the same as the classical P1-RT0-P0 discretization. Based on the well-posedness of the eliminated system, we design block preconditioners that are parameter robust. Numerical experiments are presented to confirm the theoretical results of the stabilized scheme as well as the block preconditioners. We will also present some other works on poroelastic problem briefly.

芮洪兴，教授，博士生导师。主要研究偏微分方程数值解法、科学与工程计算、油水资源数值方法及应用等。在计算数学顶级期刊Numer Math、SIAM J Numer Anal、Math Comp，流体力学顶级期刊J Fluid Mech等发表多篇高水平论文。承担国家自然科学基金重点项目、面上项目、中石化石油勘探院开发研究院外协国家重大科技专项课题等。曾获教育部自然科学一等奖、山东省自然科学二等奖（首位）等。现任中国工业与应用数学学会常务理事、山东省数学会计算数学专委会主任。