报告题目：On the use of HDiv Spaces in Computational Fluid Mechanics

报 告 人：Prof. Philippe Devloo（State University of Campinas巴西坎皮纳斯州立大学）

报告时间：2024年10月28日（星期二）9：30

报告地点：数学科学学院114（小报告厅）

邀 请 人：李崇君 教授

报告摘要：In this talk, the research on the development and use of Hdiv approximation spaces is discussed when applied to the area of computational fluid mechanics. Hdiv spaces are crucial in the construction of consistent and stable mixed finite element approximation. In a first part a constructive way for generating HDiv compatible spaces is presented based on the multiplication of a vector field by scalar shape functions. In the context of De Rham compatible sequences of spaces, Hdiv can be optimized to create approximation spaces whose divergence space is strictly determined.

The system of equations of mixed finite element approximation are a saddle point problem. An iterative method is developed where the saddle point problem is preconditioned by the inversion of the lowest order model. The preconditioner is optimal in that the resulting system can be accelerated using the conjugate gradient method, the approximation results in a conservative solution at each iteration and that the number of iterations to convergence is virtually independent of the mesh resolution and/or polynomial order.

The mixed element approximations can be extended multiscale approximations by partitioning the mesh into macro domains and meshing each macro domain by fine meshes. The resulting Multiscal Hybrid Mixed method converges both in terms of the size of the macro domain and/or fine scale mesh.

The Hdiv approximation spaces are dual to H1 approximation spaces in that the difference of an H1 and a mixed element Hdiv approximation is an effective measure of the error of either the H1 approximation or Hdiv approximation. A simple criterion is presented to, based on the estimated error, choose between h and/or p-refinement.

Finally, the feature of Hdiv approximation spaces to represent divergence free functions is explored to model incompressible Stokes or Navier Stokes flow. The continuity of the tangent velocity is satisfied weakly by a tangent traction space that acts as a Lagrange multiplier. Optimal orders of convergence are obtained and numerically confirmed.

Applying a second hybridization to the tangent traction leads to formulation where, using static condensation, the Stokes problem is modeled by a positive definite system of equation with a single constraint associated with each element.

报告人简介：Prof. Philippe Devloo is a full professor at the school of civil engineering, architecture and urbanism of the university of UNICAMP. He obtained his PhD degree in computational mechanics from the university of Texas at Austin in 1988 and has been active in the area ever since. He is a fellow from the International Association of Computational Mechanics (IACM). His research interests are in the development of efficient algorithms for the finite element method, including hp-adaptivity, error estimation, scaled boundary finite element approximations, multiscale approximations, the generation of DeRham compatible approximations spaces and others. He developed numerous research projects in collaboration with the petroleum industry studying hydraulic fracturing, wellbore reservoir interface, flow through fractured porous media, wellbore stability, core analysis and others.