MS004306-电磁场工程的泛函方法（2021秋改为MS000157泛函分析与变分法）

研究生课程开设申请表

 开

√

 课院（系、所）： 信息科学与工程学院

 课程申请开设类型： 新开□     重开□     更名□（请在□内打勾，下同）

一、课程介绍（含教学目标、教学要求等）（300字以内）

本课程介绍电磁场数值方法所涉及到的泛函分析概念和方法。让研究生用泛函概念理解并掌握变分原理、矩量法原理、加权余量法、变分法和有限元方法等等，进一步用泛函方法的统一框架，建立典型电磁场边值问题的泛函模型。了解各种数值方法的分类和比较，以及它们之间关系。

教学要求：要求学生通过课堂学习，掌握各种方法的基本原理及其应用，通过课后阅读教材和有关参考文献了解各种方法的更深入的内容和国内外的最新研究进展。

二、教学大纲（含章节目录）：（可附页）

绪论——数学物理方程（静电场方程）的经典解和广义解

§1 经典解

§2 广义解

§3 数值解

1. Lebesgue测度和Lebesgue积分

§1 Lebesgue测度

 一．Lebesgue 测度

2. Lebesgue可测集

§2 Lebesgue积分

1. 可测函数

2. Lebesgue积分

2. 度量空间和赋范线性空间

§1 度量空间

§2 赋范线性空间

1. 线性空间

2. 线性空间上的范数

3. L^p空间

§3 度量空间的完备性及Banach空间

1. 完备空间及性质

2. 度量空间的完备化

3. 稠密性（稠密性概念、可析点集）

4. Banach空间

§4 不动点原理

3. 有界算子与泛函

§1 有界线性算子

1. 线性算子与线性泛函概念

2. 线性算子的有界性与连续性

3. 线性算子空间

§2有界线性泛函与共轭空间

1. 有界线性泛函

2. 共轭空间与连续线性泛函延拓

3. 二次共轭空间与共轭算子

4. Hilbert 空间

§1 内积

1. 内积与内积空间

2. 内积与范数

§2 Hilbert空间

1. Hilbert空间

2. 内积与范数

3. 正交与投影

4. 最小二乘法

§3标准正交系与Fourier展开

1. 标准正交系

2. 正交化方法

3. Fourier级数

4. 完备的标准正交基及其性质

§4有界线性泛函与共轭空间

1. 有界线性泛函及其表示（Riesz泛函表示）

2. 共轭空间与自共轭空间

3. 共轭算子

§5 双线性Hermite泛函与自共轭算子

5. 广义函数与索伯列夫（Sobolev）空间

§1  基本函数与广义函数

§2  广义函数的导数

§3  广义函数的Fourier变换

§4  Sobolev空间

§5  实指数Sobolev空间

§6  嵌入定理

§7  迹定理

6. 电磁场问题的变分方法

§1  变分法原理及基本概念  

1.  函数的变分

2. 泛函的变分

3. 泛函极值

§2  Euler方程与奥氏方程

§3  微分方程的变分形式

§4  变分形式与双线性泛函（内积）

§5  有限维空间上的近似解与精确解的关系

7. 电磁场问题的数值方法

§1  电磁场问题的泛函模型

§2*  矩量法

§3  加权余量法

§4  Ritz方法

§5  Galerkin方法

§6  矢量有限元方法（Edge Element)

§7* 有限元方法

 注：星号部分为选讲内容。

三、教学周历

四、主讲教师简介：

孙连友：1990年7月毕业于复旦大学数学研究所，获硕士学位。2003年7月于东南大学无线电工程系获博士学位。1990年到1998年1月期间在东南大学数学系工作，从事教学和应用数学方面的研究。2005年8月至2006年8月由国家留学基金委公派到加拿大McGill大学从事博士后研究。目前，主要从事微波毫米波理论与技术、计算电磁学数值理论和计算方法方面的教学和科研工作。在IEEE和国内核心期刊及国际会议上发表论文十余篇。并合著有《电磁场边值问题的区域分解算法》，由科学出版社出版。

五、任课教师信息（包括主讲教师）：

Application Form For Opening Graduate Courses

S

√

 chool (Department/Institute)：

Course Type: New Open □   Reopen □   Rename □（Please tick in □, the same below）

1. Course Introduction (including teaching goals and requirements) within 300 words:

The course mainly expatiates on the functional concepts and numerical methods applied to electromagnetical field problems. Students need understand and use variational principle, moment methods and weighted residual methods etc. to model the electromagnetical field problems. Furthermore, students need compare them to each other, and understand them relationships.

2. Teaching Syllabus (including the content of chapters and sections. A sheet can be attached):

Introduction---Classical and Generalized Solutions of Mathematical Physics Equations

§1 Classical Solutions

§2Distributions

§3Numerical Solutions

Chapter 1. Lebesgue Measure & Lebesgue Integral

§1 Lebesgue Messure

1. Lebesgue Measure

2.Lebesgue Measurable Set

§2 Lebesgue Integral

1.Measurable function

2. Lebesgue Integral

Chapter2.  Metric Space & Normed Linear Space

§1 Metric Space

§2 Normed Linear Space

1. Linear Space

2. Norm on Linear Space

3. L^p Space

§3 Completeness of Metric Space and Banach Space

1. Completed Space and its Properties

2. Completion of Metric Space

3. Density(Concept of Density, Separable Metric Space)

4. Banach Space

§4 Fixed Point Theory

Chaper 3 Bounded Operator and Functional

§2 bounded Linear Operators

1. Concepts of Linear Operator and Linear Functional

2. Continuation and bound of Linear Operators

3. Space of Linear Operators

§3 Bounded Linear Functional and Conjugate Space

1. Bounded Linear Functional

2. Conjugate Spaceand Prolongation of Linear Continuous Functional

3. Quadratic Conjugate Space and Conjugate Operator

Chaper 4Hilbert Space

§1 Inner Product

1. Inner Product and Its Space

2. Inner Product and Norm

3. Space of Linear Operators

§2 Hilbert Space

1. Hilbert Space

2. Inner Product and Norm

3. Orthogonal Vector and Its Projection

4. Method of Least Squares

§3 Normal Orthogonal System and Fourier Expansion

1. Normal Orthogonal System

2. Methods of Orthogonalization

3. Fourier Series

4. Completed Normal Orthogonal System and Its Properties

§4 Bounded Linear Functional and Its Conjugate Space

1. Bounded Linear Functional and Its Representation

2. Conjugate Space and Self- Conjugate Space

3. Conjugate Operator

§5 Bilinear Hermite Functional and Self-Conjugate Operator

Chapter 5 Distribution and Sobolev Space

§1 Basic Function and Distribution

§2 Derivatives of Distribution

§3 Fourier Transformation of Distribution

§4 Sobolev Space

§5Sobolev Space With Real Exponent

§6 Embedding Theorem

§7 Trace Theorem

Chapter 6 Variational Method for Electromagnetical Field

§1 Variational Principle and Its Basic Concepts

1. Variational of Function

2. Variational of Functional

3. Extremum of Functional

§2 Euler Equations and OCTPOГPaДCКИЙEquations

§3 Variational Formula of Differential Equations

§4 Variational Formula and Bilinear Functional

§5Relationship of Solutions Between Finite Dimensional Space and Infinite Dimensional Space

Chapter 7 Numerical Methods of Electromagnetical Field Problems

§1*. Moment Methods

§2. Weighted Residual Method

§3. Ritz Method

§4.Galerkin Method

§5. Edge Element Method

§6*. Finite Element Method

3. Teaching Schedule:

Note: 1.Above one, two, and three items are used as teaching Syllabus in Chinese and announced on the Chinese website of Graduate School. The four and five items are preserved in Graduate School.

2. Course terms: Spring, Autumn , and Spring-Autumn term.  

3. The teaching languages for courses: Chinese, English or Chinese-English.

4. Applicable range of discipline: public, first-class discipline, second-class discipline, and third-class discipline.

5. Practice includes: experiment, investigation, research report, etc.

6. Teaching methods: lecture, seminar, practice, etc.

7. Examination for degree courses must be in paper.

8. Teaching material websites are those which have already been announced.

9. Brief introduction of chief lecturer should include: personal information (date of birth, gender, degree achieved, professional title), research direction, teaching and research achievements. (within 100-500 words)

4. Brief Introduction of Chief lecturer:

Lianyou Sun received M.Sc. degree in applied mathematics from Fudan University, Shanghai, China, in 1990. PhD degree in electromagnetic field and microwave technology from Southeast University, Nanjing, China, in 2003.

He worked at Southeast University, Nanjing, China, firstly in Department of Applied Mathematics as a lecturer from 1990 to 1997, and then in Department of Radio Engineering as a associate professor from 1998 to 2005. With Financial support of China Scholarship Council, he pursued his postdoctoral studies in computational electromagnetics at McGill University, Montreal, Canada from Aug. 2005 to Aug. 2006. His current interests are in numerical methods for electromagnetic problems, especially for large scale problems. He also is a co-authorof a book:Domain Decomposition Methods for Solving the Electromagnetic field Boundary Value Problems,Beijing, China: Science Press, 2005. He has published more than ten papers on computational electromagnetics.

5. Lecturer Information (include chief lecturer)