报告题目：Local analysis of the local discontinuous Galerkin method with generalized fluxes for nonlinear singularly perturbed convection-diffusion-reaction problems

报告人：孟雄，哈尔滨工业大学

报告时间：5月21日，周四，上午10:00-11:00 

报告地点：五教5505

摘要：

In this talk, we analyze the local discontinuous Galerkin (LDG) method with generalized numerical fluxes for time-dependent nonlinear singularly perturbed convection-diffusion-reaction problems with outflow boundary layers in one dimension. The use of generalized numerical fluxes produces high resolution without visible oscillations in comparison with classical upwind and alternating numerical fluxes, when two-phase flows with gravitational effects are concerned. Local analysis is carried out by introducing the weight function $\psi$ into the L$^2$-norm error estimates. Two equivalent weighted norms are introduced to handle the nonlinear convection term with a positive lower bound, and the properties of the LDG spatial discretization are established. By extending the standard boundedness of the L$^2$ projection and the weighted inverse inequalities from $\psi^{-1}$ weight to $\psi^{-2}$ weight, and using the stability result, we derive a sharp bound for high order terms in the Taylor expansion of $f(u)$. Then, under a local a priori assumption, optimal error estimates are proved in a local region. Numerical experiments confirm the validity of theoretical results, illustrate long time behaviors, and show the efficiency for the fully degenerate nonlinear Buckley-Leverett problems.

报告人简介：

孟雄，哈尔滨工业大学数学学院教授、博导，欧盟玛丽居里学者、美国布朗大学访问学者，主要研究方向为对流占优偏微分方程的高阶精度数值方法，尤其是间断有限元方法的构造、分析与应用。在SIAM Journal on Numerical Analysis, Numerische Mathematik, Mathematics of Computation, Journal of Computational Physics等计算数学期刊发表论文20余篇。主持欧盟“玛丽居里行动”计划基金、国家自然科学基金面上项目、国家自然科学基金青年基金等项目。获龙江学者青年学者（2025）、CSIAM应用数学青年科技奖（2022）、国家天元东北中心优秀青年学者（2021）等奖励。