﻿2021年数学学院“吉大学子全球胜任力提升计划”研究生系列短课程（17）-吉林大学数学学院

# 2021年数学学院“吉大学子全球胜任力提升计划”研究生系列短课程（17）

Abstract: The course goal is to introduce finite element methods (FEM) to solve a number of partial differential equations such as Poisson equation, heat equation and Navier-Stokes. An in-depth knowledge on FEM as well as iterative solvers such as multigrid methods will be discussed in a platform of FEniCS. The course will be maintained to provide not only algorithmic techniques but also a hands-on experience to implement the methods. After students complete the course works, they are expected to have abilities to tackle a number of partial differential equations using FEniCS.

 授课日期Date of Lecture 课程名称（讲座题目）Name (Title) of Lecture 授课时间Duration (Beijing Time) 参与人数Number of Participants 2021.07.27 Introduction to Finite   Element Methods (FEMs) 3:30pm-4:30pm 15-20 2021.07.29 FEMs for Elliptic PDEs 3:30pm-4:30pm 15-20 2021.08.03 FEMs for Parabolic PDEs 3:30pm-4:30pm 15-20 2021.08.05 Mixed FEMs for Poisson   Equation (Eq) 3:30pm-4:30pm 15-20 2021.08.10 Mixed FEMs for Stokes   Eqs 3:30pm-4:30pm 15-20 2021.08.12 Mixed FEMs for   Navier-Stokes Eqs 3:30pm-4:30pm 15-20

Lecture 1: We introduce a couple of physical problems, and a list of partial differential equations that will be discussed in the total of six lectures. Some basic information about finite element methods, Sobolev spaces, and FEniCS will be discussed.

Lecture 2: We study the conforming finite elements and how to apply them to solve Poisson equation. We also discuss and demonstrate their implementations using FEniCS.

Lecture 3: We extend the Lecture 2 to use the conforming finite elements to solve heat equation. We also discuss and demonstrate their implementations using FEniCS.

Lecture 4: We formulate the Poisson equation in a mixed formulation and discuss why such a formulation is useful. We study Raviart-Thomas and BDM finite elements to approximate vector unknowns. We then apply this to solve the mixed formulation of Poisson equation.

Lecture 5: We study the basic fluids dynamics equation, called the Stokes equation. This consists of two unknowns, velocity and pressure. We study how to solve this problem as an extension of Lecture 4. We then apply this to solve the Stokes equation in FEniCS framework.

Lecture 6: We discuss the Navier-Stokes equation, a full equation for Newtonian fluids motion. We first study how to handle the nonlinear term using fixed point iteration and then will solve the Navier-Stokes equation in FEniCS framework.