11月20日上午

9:30-10:20

程晓良，浙江大学

题目：Numerical analysis for variational and Hemi-variational inequality

摘要：This talk consists two parts: For the variational inequalities model obstacle problems, we propose and analyze some numerical algorithms based on some equivalent formulations. The algorithms for the related discrete Hamilton-Jacobi-Bellman equation are also discussed.The second part concerns a Quasistatic history-dependent contact model. It is a hemi-variational inequalities model. We analyze the finite element methods to approximate it, derive some error estimates and analyze some algorithms. Several models for different contact conditions in hemi-variational inequalities are mentioned.

10:20-11:10

王飞，西安交通大学

题目：Virtual Element Method for General Elliptic Hemivariational Inequalities

摘要：An abstract framework of the virtual element method is established for solving general elliptic hemivariational inequalities with or without constraint, and a unified a priori error analysis is given for both cases. Then, virtual element methods of arbitrary order are applied to solve three elliptic hemivariational inequalities arising in contact mechanics, and optimal order error estimates are shown with the linear virtual element solutions.  Numerical simulation results are reported on several contact problems; in particular, the numerical convergence orders of the lowest order virtual element solutions are shown to be in good agreement with the theoretical predictions. This is a joint work with Bangmin Wu and Weimin Han.

11:10-12:00

袁达明，江西师范大学

题目：Positivity-preserving linear discontinuous scheme for solving neutron transport equations in heterogeneous slabs

摘要：For the numerical solution of neutron transport equations with spatially discontinuous varying cross sections, the positivity-preserving properties are important and challenging issues. In particular, for a heterogeneous slab, in the regions of zero total cross section and source free cells, the linear scaling limiter may not be valid when both the cell average and the cell edge fluxes are negative. In this work, we propose two limiters, called ``inflow preferred'' and ``average preferred,'' to cope with this ``worst case'' situation. These limiters can maintain the original polynomial accuracy. We discretize the spatial variable by employing the standard linear discontinuous scheme and linearizing the external source term.

Numerical results for solving Reed-like problems are provided to verify the efficiency of the proposed schemes and to compare with other positivity-preserving techniques. The two proposed limiters are more optimal in the sense of introducing less conservation error.

11月20日下午

14:00-14:50

龚荣芳，南京航空航天大学

题目：Second order asymptotical regularization methods for PDEs based inverse source problems

摘要：This talk considers an inverse source problem of PDEs with both Dirichlet and Neumann boundary data. Unlike the existing methods, which usually employ the first-order in time gradient-like system for numerically solving the regularized optimization problem with a fixed regularization parameter, we propose a new method with a second-order in time dissipative gradient-like system and a dynamical selected regularization parameter. A damped symplectic scheme is proposed for the numerical solution. Furthermore, we develop a second order asymptotical regularization method for the purpose of acceleration. Theoretical analysis is given for both the continuous models and the numerical algorithms. Several numerical examples are given to assess the behavior of the proposed methods.

14:50-15:40

朱升峰，华东师范大学

题目： Numerical approximations of shape gradients in shape optimization

摘要： Eulerian derivatives of shape functionals in optimal shape design can be written in two formulations: boundary and volume integrals. The former is widely used in shape gradient descent algorithms. The latter holds more generally, although rarely being used numerically in literature. We consider finite element approximations to the two types of shape gradients from corresponding Eulerian derivatives. We present convergence analysis with a priori error estimates, which show that the volume integral formula converges faster and offers higher accuracy. Numerical results with applications in shape optimization are presented.

15:40-16:30

周圣高，苏州大学

题目：Physical-property preserving numerical methods for the Poisson-Nernst-Planck equations

摘要： The Poisson-Nernst-Planck (PNP) type of equations are one of the most extensively studied models for the transport of charged particles in many physical and biological problems. The solution to the PNP equations has many properties of physical importance, e.g., positivity, mass conservation, energy dissipation. It is desirable to design numerical methods that are able to preserve such properties at discrete level. In this talk, we will present some recent advances on the development of numerical methods that can maintain physical properties. Some numerical results are shown to demonstrate their performances.