北京雁栖湖应用数学研究院

This talk will firstly present nonconforming finite element spaces for $\mathrm{H}\Lambda\mathrm{k}$ developed through the inheritance of adjoint relations between the operators $\mathbf{d}$ and $\mathbf{\delta}$. Finite element spaces for $\mathrm{H}\Lambda\mathrm{k} \mathrm{\cap} \mathrm{H}^{\ast} _ 0 \Lambda\mathrm{k}$ , constructed by following a similar design principle with the nonconforming playing a crucial role, are then presented, for which uniform discrete Poincar $\mathrm{e}$ inequalities are established. Based on these constructions, primal finite element schemes for the Hodge Laplace equations are formulated, and their validities are verified through numerical experiments.

This talk is concerned with the analysis of a class of spectral volume (SV) methods for hyperbolic equations. We first prove that for a general nonuniform mesh and any polynomial degree $k$, the proposed SV methods are stable and can achieve optimal convergence orders in the $L^2$ norm. Secondly, we prove that the SV methods have some superconvergence properties at some special points. Moreover, we demonstrate that for constant-coefficient equations, the RRSV method is identical to the upwind discontinuous Galerkin (DG) method. Our theoretical findings are validated with several numerical experiments at the end.

The numerical approximation of low-regularity solutions to the nonlinear Schrodinger equation (NLSE) is notoriously difficult and even more so if structure-preserving schemes are sought. Recent works have been successful in establishing symmetric low-regularity integrators for NLSE. However, so far, all prior symmetric low-regularity algorithms are fully implicit, and therefore require the solution of a nonlinear equation at each time step, leading to significant numerical cost in the iteration. In this work, we introduce the first fully explicit (multi-step) symmetric low-regularity integrators for NLSE. We demonstrate the construction of an entire class of such schemes which notably can be used to symmetrise (in explicit form) a large amount of existing low-regularity integrators. We provide rigorous convergence analysis of our schemes and numerical examples demonstrating both the favourable structure preservation properties obtained with our novel schemes, and the significant reduction in computational cost over implicit methods.

For incompressible Navier-Stokes (NS) equation and Cahn-Hilliard-NS two phase flow, we propose the dynamically regularized Lagrange multiplier methods (DRLM). Via the Lagrange multiplier, we incorporate the energy evolution process into the original system. The corresponding first- and second-order DRLMs are unconditionally energy stable when the nonlinear terms are treated explicitly. In addition, we introduce a dynamically regularized term, i.e., the stabilized first-order time derivative on the square of the Lagrange multiplier, so that the resulting numerical schemes allow for a larger time step. Lastly, we provide the temporal semi- and fully-discretization error estimates on the first-order DRLM scheme.

High mesh quality plays a crucial role in maintaining the stability of solutions in geometric flow problems. In this work, we employ the minimal deformation (MD) formulation for anisotropic surface diffusion to introduce an artificial tangential velocity determined via harmonic mapping, thereby improving mesh quality in the numerical simulation. Furthermore, we develop a class of structure-preserving algorithms for the resulting MD-based flows, including both first-order and high-order temporal discretization schemes. Extensive numerical experiments show that our methods effectively preserve mesh quality while achieving high-order temporal accuracy, as well as volume conservation and/or energy stability.

在本次报告中，我将介绍一种采用调和平均技术的杂交间断伽辽金（Hybridizable DG）格式，用于求解半导体器件耦合外电路的漂移扩散模型。该格式结合了调和平均的稳健性，以及杂交间断伽辽金格式所具备的高阶精度与hp自适应灵活性。我们通过引入基于梯度的指标以实现hp自适应细化，采用自适应步长的隐式时间方法保证计算的稳定性与精度。数值算例表明，该格式即便应用于高掺杂及突变PN结场景，也不会出现数值振荡现象。

Gross-Pitaevskii 型方程是研究超冷量子气体的重要模型。为刻画平均场近似之外的物理效应，人们发展了多种扩展模型，如含 Lee-Huang-Yang 修正的扩展 Gross-Pitaevskii 模型和 unitary Fermi gas有效模型。此类模型不仅具有更复杂的非线性结构，也给基态问题的理论分析与数值计算带来了新的挑战。本报告将介绍我近年来在相关模型上的工作，重点讨论上述两类模型中的基态问题，包括若干存在性结果，以及相应的数值算法与计算结果。

The Dirac equation is a relativistic wave equation which plays an important role in relativistic quantum physics and provides a natural description of relativistic spin-1/2 particles. In this talk, we present numerical methods including several finite difference methods, the symmetric and asymmetric exponential wave integrator Fourier pseudospectral methods and establish the error estimates for the discretization of the Dirac equation in different regimes. Extensive numerical results are reported to support our error estimates.