报告题目: Stabilizations for convection-dominated equations
摘要: For convection-dominated problems arising from computational fluid dynamics, a typical difficulty of the finite element method of this type of problems of convection-dominated is that the finite element solution in not-too-fine meshes is usually globally highly oscillatory, due to too small diffusivity and too large dissipative reaction term. In practice, physical and chemical processes would involve such situations, often seen in nearly turbulent flows and fast chemical reaction and explosion and combustions. Mathematically, the solution exhibits fast changing behavior and rather large gradient, with abruptly drops in values of the solution, in some narrow sub-regions, particularly in the immediate vicinity of the boundary. Such sub-regions are referred to as boundary and interior layers. Although, in theory, a sufficiently fine mesh could result in a convergent finite element solution, the resultant approximation has very low resolution and is even so highly distorted that is not practically useful at all. An efficient strategy for dealing with this difficulty for convection-dominated problems is the so-called stabilized finite element method. This method features some stabilizations which are obtained by locally solving the original partial differential equations in the convection-dominated problems so that the unresolved component of the solution in the boundary and interior layers could be incorporated into the stabilization terms. Consequently, an accurate finite element solution with not-too-fine meshes can be expected. In this talk I will present new stabilization finite element methods. Theoretical results and numerical results are included in my talk to justify the efficiency and the effectiveness of the new methods.
报告人简介：段火元，武汉大学数学与统计学院教授，博士生导师。研究兴趣：偏微分方程数值解、有限元方法、多重网格算法、自适应算法、预处理迭代算法；随机微分方程的数值方法及应用；图像处理的数值方法；反问题数值方法。在国内外学术期刊发表学术论文近50篇，其中，在SIAM Journal On Numerical Analysis，Mathematics of Computation，Numerische Mathematik，Computer Methods in Applied Mechanics and Engineering，IMA Journal of Numerical Analysis，Journal of Computational Physics等国际著名的计算数学专业期刊发表学术论文20篇。