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本课程是新加坡-麻省理工学院联盟(SMA)的部分课程内容，将在麻省理工学院由该校教师对该校学生讲授，并同时以视讯方式播送至国立新加坡大学(NUS)，给该校SMA学院的学生学习。某些情况下授课方式可以相反：课程内容将在NUS由该校SMA学院的教师讲授，并以视讯方式播送至麻省理工学院。
MIT学分：3-0-9，高等研究生学分。
课程内容
本课程讲授求解不同线性及非线性椭圆、抛物线及双曲线偏微分方程式与积分方程式等之现代数值技巧基础，并强调在许多科学、工程及相关领域上的应用。课程主题有：数学列式、有限差分与有限体积离散、有限元素离散、边界元素离散、直接与叠代解法。
先修课程
麻省理工学院之18.03或18.06或同级课程，并须熟习MATLAB®。
授课教师
Jaime Peraire 教授
Anthony T. Patera 教授
Jacob White 教授
Boo Cheong Khoo 教授
授课资料
主要资料:
上课讲义：在课前提供；列档之教师授课影片：在课后提供。
参考书籍
(1) 《数值线性代数》(Numerical Linear Algebra), L. N. Trefethen 与 D. Bau, III, SIAM.
(2) 《守恒律之数值方法》(Numerical Methods for Conservation Laws,), R. Levecque, 数学教学, ETH Zurich, Birkhauser.
(3) 《有限元素法分析》(Analysis of the Finite Element Method), G. Strang and G. J. Fix, Prentice-Hall.
(4) 《偏微分方程之数值近似法》(Numerical Approximation of Partial Differential Equations), A. Quarteroni 与 A. Valli, Springer-Verlag.
(5) 《第二类积分方程之数值解》(The Numerical Solution of Integral Equations of the Second Kind), K. E. Atkinson.
(6) 《多重格点自习指导》(A Multigrid Tutorial), W. L. Briggs, 等，第二版。 SIAM 2000.
(7) 《偏微分方程介绍：计算方法》(Introduction to Partial Differential Equations: A Computational Approach), A. Tveito 与 R. Winther, 应用数学教科书 29, Springer; 1998，本书存麻省理工学院 Barker图书馆，NUS中央图书馆及SMA图书馆。
学生评分
4组习题与专题:
有限差分：25%
双曲线方程式：20%
有限元素：25%
边界积分方程式：20%
课堂互动：10%
MATLAB®是Mathworks公司之注册商标。
This course is offered as part of the Singapore-MIT Alliance (SMA), and will be delivered at MIT (by MIT faculty) for MIT students and simultaneously broadcast to the National University of Singapore (NUS) for SMA students. In some cases the roles will reverse: the classes will be delivered at NUS (by NUS SMA faculty) and simultaneously broadcast to MIT.
MIT Units: 3-0-9, Graduate H-level Credit
Description
A presentation of the fundamentals of modern numerical techniques for a wide range of linear and nonlinear elliptic, parabolic and hyperbolic partial differential equations and integral equations central to a wide variety of applications in science, engineering, and other fields. Topics include: mathematical formulations, finite difference and finite volume discretizations, finite element discretizations, boundary element discretizations, direct and iterative solution methods.
Prerequisites
MIT: 18.03 or 18.06 or equivalent; and familiarity with MATLAB®.
Instructors
Prof. Jaime Peraire
Prof. Anthony T. Patera
Prof. Jacob White
Prof. Boo Cheong Khoo
Course Materials
Primary Source:
Lecture notes are available before class, and archived lecture videos are available after class.
Reference Texts:
(1) Numerical Linear Algebra, L. N. Trefethen and D. Bau, III, SIAM.
(2) Numerical Methods for Conservation Laws, R. Levecque, Lectures in Mathematics, ETH Zurich, Birkhauser.
(3) Analysis of the Finite Element Method, G. Strang and G. J. Fix, Prentice-Hall.
(4) Numerical Approximation of Partial Differential Equations, A. Quarteroni and A. Valli, Springer-Verlag.
(5) The Numerical Solution of Integral Equations of the Second Kind, K. E. Atkinson.
(6) A Multigrid Tutorial, W. L. Briggs, et al. 2nd ed. SIAM 2000.
(7) Introduction to Partial Differential Equations: A Computational Approach, A. Tveito and R. Winther, Texts in Applied Mathematics 29, Springer; 1998, are on reserve at MIT's Barker Library and at NUS's Central Library and SMA Library.
Assessment
4 Problem Sets/Projects:
Finite Differences: 25%
Hyperbolic Equations: 20%
Finite Elements: 25%
Boundary Integral Eqs.: 20%
Class Interaction 10%
MATLAB® is a trademark of The MathWorks, Inc.
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