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Workshop on Numerical Analysis

编辑:wfy 时间:2018年10月12日 访问次数:893

浙江大学数学科学学院九十周年院庆系列活动之八十三


时间:2018年10月13日 13:00-17:40

地点:工商楼200-9

联系人:程晓良xiaoliangchen@zju.edu.cn,梁克维matlkw@zju.edu.cn


报告1(13:00-13:50)

报告题目:Second order asymptotical regularization methods for inverse problems in partial differential equations

报告人:龚荣芳(南京航空航天大学)

摘要:We develop Second Order Asymptotical Regularization (SOAR) methods for solving inverse source problems in elliptic partial differential equations with both Dirichlet and Neumann boundary data. We show the convergence results of SOAR with the fixed damping parameter, as well as with a dynamic damping parameter, which is a continuous analog of Nesterov's acceleration method. Moreover, by using Morozov's discrepancy principle together with a newly developed total energy discrepancy principle, we prove that the approximate solution of SOAR weakly converges to an exact source function as the measurement noise goes to zero. A damped symplectic scheme, combined with the finite element method, is developed for the numerical implementation of SOAR, which yields a novel iterative regularization scheme for solving inverse source problems. Several numerical examples are given to show the accuracy and the acceleration effect of SOAR. A comparison with the state-of-the-art methods is also provided.


报告2(13:50-14:40)

报告题目:A Unified Study of Continuous and Discontinuous Galerkin Methods

报告人:王飞(西安交通大学
摘要:During the past three decades, many different discontinuous finite element methods (FEMs), including discontinuous Galerkin (DG) methods, hybrid DG methods and weak Galerkin methods, have been developed for solving a wide range of partial differential equations. In this talk, taking Poisson problem as an example we present a general framework for understanding these discontinuous finite element methods by the concept of DG-derivatives. In light of this general framework, a new mixed DG method is proposed, and we apply it to solve linear elasticity problem with arbitrary order discontinuous finite element spaces.


报告3(14:40-15:30)

报告题目:Robust numerical methods for solving the neutron transport equations

报告人:袁达明(江西师范大学
摘要:For the numerical solution of neutron transport equations, both the positivity-preserving property and the speed up of the iterative convergence are important and challenging issues. In this presentation, a high-order positivity-preserving scheme is presented firstly. And then the combination of S2 synthetic acceleration method and positivity-preserving scheme are derived to solving Reed-like problem. Numerical results are provided to verify the efficient of the schemes.


报告4(16:00-16:50)

报告题目:Positivity preserving, conservative, and free energy dissipating finite difference methods for multi-dimensional PNP-type equations

报告人:周圣高(苏州大学)

摘要:computational investigations are performed to study condition numbers of the semi-implicit discretization of the NP equations, further revealing advantages of the scheme in computational efficiency and stability. Our estimates on the upper bound of condition numbers indicate that the developed discretization based on harmonic-mean approximations can effectively solve a known issue---a large condition number is often accompanied by the use of Slotboom variables. Numerical tests verify that the numerical solution respects desired properties, and is second-order accurate in space and first-order accurate in time. An application of the numerical scheme to an electrochemical charging system demonstrates its effectiveness in solving realistic problems.

This is a joint work with Jie Ding, Yiran Qian, and Zhongming Wang.


报告5(16:50-17:40

报告题目:Petrov-Galerkin and indirect finite element methods for space-fractional diffusion equations with variable-coefficients

报告人:朱升峰(华东师范大学

摘要:Fractional diffusion equations have found increasingly more applications in recent years but introduce new mathematical and numerical difficulties. Galerkin formulation, which was proved to be well-posed for fractional diffusion equations with a constant diffusivity coefficient, may lose its coercivity for variable-coefficient problems. The corresponding finite element method fails to converge. We develop a Petrov-Galerkin finite element method for variable-coefficient fractional diffusion equations. We prove the well-posedness and optimal-order convergence. Moreover, we present an indirect finite element method, which reduces the solution of fractional diffusion equations to that of second-order diffusion equations post-processed by a fractional differentiation. It reduces numerically the computational work and the memory requirement. We prove that the corresponding high-order methods achieve high-order convergence rates even though the true solutions are not smooth. Numerical results are presented.