Few days ago, Teacher Wen Hairui and collaborators of School of Mathematics and Statistics, Beijing Institute of Technology(BIT) published the title of "Error estimates for the AEDG method to one" in the top international academic journals "Math. Comp." And "Arch. -dimensional linear convection-diffusion equations "and a research paper entitled" Global structure and regularity of solutions to Eikonal equation ".

　　The Hamilton-Jacobi (HJ) equation was first proposed in classical mechanics, and it has extensive and profound applications in dynamic programming, variational calculation, and optimal control. The numerical simulation and theoretical analysis of such equations have always been a research hotspot. In numerical calculation, the design of conservative, compatible and high-precision numerical methods is the focus of people‘s attention. In terms of theoretical analysis, the research on the properties of viscous solutions for superlinear, strictly convex Hamiltonian equations is relatively complete, but there is almost no theoretical analysis for the geometric Hamiltonian HJ equations.

　　AEDG method was first proposed by Iowa State Univ. Prof. Liu Hailiang and postdoc M. Pollack. This method does not require numerical flux and is compatible and conservative. For linear convection-diffusion equations, the stability of AEDG is proved under CFL-type conditions.

　　Based on that, Teacher Wen Hairui and Prof. Liu Hailiang creatively introduced two approximation spaces, corresponding bilinear operators and coupled global projections, therefore overcame the difficulties caused by the polynomial coincidence in the alternating development system. The dual method often used in classical finite element error estimation is redesigned, so that the energy error of the semi-discrete AEDG format is successfully raised to the optimal square error. For spatial dispersion, using any higher-order polynomial, a fully discrete scheme using second-order Crank-Nicolson and third-order Runge-Kutta time dispersion is designed. At the same time, under the condition of CFL stability, they got the corresponding optimal square error order.

　　In addition to the HJ equation, the AEDG method has achieved a good approximation effect in the conservation law and convection-diffusion equation calculation. Not requiring numerical flow characteristics, it overcomes the difficulty of designing a stable flow rate in the general DG method. The results of the article and the new proof method lay the theoretical foundation for the convergence of the AEDG method, which is of great significance.

　　In terms of the regularity theory of the HJ equation, for the Eikonal equation, since the restriction on the singular point is weaker, the singular point does not necessarily have the same regularity as the initial value and other characteristics, which makes the situation extremely complicated. At the same time, its non-smooth, non-strictly convex Hamiltonian also brings great challenges to theoretical analysis.

　　Teacher Wen Hairui along with collaborators used sub-differentials to define the characteristic line at non-differentiable points, and at the same time, creatively introduced effective features and feature termination time.For the initial values of C1 and C2, the definition of singular point set, the countability of connected branches, the regularity and global structure of solutions outside the singular point set are given. The reviewers evaluated the results to make up for the theoretical gap of the properties analysis of the solution of this special HJ equation.

Links to the paper:

https://www.ams.org/journals/mcom/2018-87-309/S0025-5718-2017-03226-9/home.html

https://www.esaim-m2an.org/articles/m2an/abs/2018/05/m2an170163/m2an170163.html

https://link.springer.com/article/10.1007/s00205-018-01339-4 Attached

personal profile:

　　Wen Hairui, lecturer, Ph.D. from Institute of mathematics and systems science, Chinese Academy of Sciences, and postdoctoral at the Zhou Peiyuan Center for Applied Mathematics, Tsinghua University Engaged in the regularity of Hamilton-Jacobi equation, the large time step method of conservation law, the theoretical analysis and numerical simulation research of DG and structure-preserving DG methods, his nine papers were published in the journals such as Arch. Ration. Mech. Anal., Math.Comp.and so on.

Contribution:School of Mathematics and Statistics
Editor:Tao Siyuan Audit: Tian Yubin

Translator:Zhao Shengtao, News Agency of BIT