Spring 2013:
Numerical Methods for Partial Differential Equations (Graduate Course)
Class Information:
Office Hour: Tuesday afternoon, 2:00-4:00
Office Location: 624 Pao Yue-Kong Library
email: lzhang2012 AT sjtu
TA Office Hour: Tuesday morning, 8:30-11:00
TA Office Location: 数学系博士办公室（包玉刚图书馆背面数学系三楼）
email: zhouanwa AT 126.com
Homework out: Each Friday
Homework due: Every two weeks, Thursdays in even weeks
Homework 20%
Projects 30%
Midterm 20%
Final 30%
Homework:
Homework 1, Homework 2, Homework 3, Homework 4, Homework 5, Homework 6, Homework 7, Homework 8,
Homework 9: Ying Long'an book, Chapter 1, Problem 5 (p. 28), Chapter 2, Problem 1,2, (p. 62).
Homework 10: Ying Long'an book, Chapter 3, Problem 1,2 (p. 74)
Project Topics:
1. The (unfinished) PDE Coffee Table Book
by Lloyd N. Trefethen and Kristine Embree
2. MIT Open Course, Mathematical Methods for Engineers II
by Gilbert Strang
3. Your future research topic
Exams:
Example Code:
Syllabus:
Week 1:
Solve Poisson Equation:
Finite difference method, formulation of 5-point method, linear system.
Readings:
Iserles 147-151.
Week 2:
Solve Poisson Euqation:
Finite difference method, existence and uniqueness of the solution, error estimate of the solution, higher order method, treatment of curved boundary.
Finite Element Method for Poisson Equation: derivation of Poisson equation, solution as an energy minimizer.
Readings:
Iserles 151-156,
Notes from MIT Opencourse website, Finite Element Methods for Elliptic Problems; Variational Formulation: The Poisson Problem (PDF),
Week 3:
Solve Ax = b:
LU factorization for sparse banded matrix, operation count.
Iterative method: linear one-step stationary scheme, necessary and sufficient condition for convergence, Jacobi iteration, Gauss-Seidel iteration, eigensystem for TST matrix, convergence
rate of Jacobi and G-S methods for TST matrix.
Readings: Iserles 233-243, 252-260, 264-267.
Week 4:
Solve Ax=b:
Steepest descent and conjugate gradient
Readings: Iserles 309-317.
A more 'transparent' illustration of conjugate gradient can be found at http://www.cs.cmu.edu/~quake-papers/painless-conjugate-gradient.pdf (An Introduction to
the Conjugate Gradient Method Without the Agonizing Pain Edition 1.25, by Jonathan Richard Shewchuk).
Week 5:
Solving hyperbolic equations:
Review of ODE (consistency, zero-stability and convergence),
advection equaitons, boundary conditions (inflow, outflow), initital condition, characteristics,
General defintions: convergence, local truncation error, consistency, stability (Lax-Richtmyer), Lax equivalence theorem,
Readings:
LeVeque 201-205, the definition of local truncation error 183-184, the definition of convergence and stability, Lax equivalence theorem 189-190.
References:
Partial Differential Equations
A First Course in the Numerical Analysis of Differential Equations
by Arieh Iserles
Finite Difference Methods for Ordinary and Partial Differential Equations
by Randy LeVeque
The Mathematical Theory of Finite Element Methods
by Susanne C. Brenner and Larkin Ridgway Scott, available at dangdang.
有限元方法的数学基础
王烈衡，许学军编著
有限元方法讲义
应隆安
Class Information:
Office Hour: Tuesday afternoon, 2:00-4:00
Office Location: 624 Pao Yue-Kong Library
email: lzhang2012 AT sjtu
TA Office Hour: Tuesday morning, 8:30-11:00
TA Office Location: 数学系博士办公室（包玉刚图书馆背面数学系三楼）
email: zhouanwa AT 126.com
Homework out: Each Friday
Homework due: Every two weeks, Thursdays in even weeks
Homework 20%
Projects 30%
Midterm 20%
Final 30%
Homework:
Homework 1, Homework 2, Homework 3, Homework 4, Homework 5, Homework 6, Homework 7, Homework 8,
Homework 9: Ying Long'an book, Chapter 1, Problem 5 (p. 28), Chapter 2, Problem 1,2, (p. 62).
Homework 10: Ying Long'an book, Chapter 3, Problem 1,2 (p. 74)
Project Topics:
1. The (unfinished) PDE Coffee Table Book
by Lloyd N. Trefethen and Kristine Embree
2. MIT Open Course, Mathematical Methods for Engineers II
by Gilbert Strang
3. Your future research topic
Exams:
Example Code:
Syllabus:
Week 1:
Solve Poisson Equation:
Finite difference method, formulation of 5-point method, linear system.
Readings:
Iserles 147-151.
Week 2:
Solve Poisson Euqation:
Finite difference method, existence and uniqueness of the solution, error estimate of the solution, higher order method, treatment of curved boundary.
Finite Element Method for Poisson Equation: derivation of Poisson equation, solution as an energy minimizer.
Readings:
Iserles 151-156,
Notes from MIT Opencourse website, Finite Element Methods for Elliptic Problems; Variational Formulation: The Poisson Problem (PDF),
Week 3:
Solve Ax = b:
LU factorization for sparse banded matrix, operation count.
Iterative method: linear one-step stationary scheme, necessary and sufficient condition for convergence, Jacobi iteration, Gauss-Seidel iteration, eigensystem for TST matrix, convergence
rate of Jacobi and G-S methods for TST matrix.
Readings: Iserles 233-243, 252-260, 264-267.
Week 4:
Solve Ax=b:
Steepest descent and conjugate gradient
Readings: Iserles 309-317.
A more 'transparent' illustration of conjugate gradient can be found at http://www.cs.cmu.edu/~quake-papers/painless-conjugate-gradient.pdf (An Introduction to
the Conjugate Gradient Method Without the Agonizing Pain Edition 1.25, by Jonathan Richard Shewchuk).
Week 5:
Solving hyperbolic equations:
Review of ODE (consistency, zero-stability and convergence),
advection equaitons, boundary conditions (inflow, outflow), initital condition, characteristics,
General defintions: convergence, local truncation error, consistency, stability (Lax-Richtmyer), Lax equivalence theorem,
Readings:
LeVeque 201-205, the definition of local truncation error 183-184, the definition of convergence and stability, Lax equivalence theorem 189-190.
References:
Partial Differential Equations
A First Course in the Numerical Analysis of Differential Equations
by Arieh Iserles
Finite Difference Methods for Ordinary and Partial Differential Equations
by Randy LeVeque
The Mathematical Theory of Finite Element Methods
by Susanne C. Brenner and Larkin Ridgway Scott, available at dangdang.
有限元方法的数学基础
王烈衡，许学军编著
有限元方法讲义
应隆安