FSF3562 Numerical Methods for Partial Differential Equations 2022
The lectures during period 4 are held Wednesdays 13:15-15:00 in seminar room 3418.
The lectures during period 3 are held Mondays at 13:15-15:00. For the time being we use a hybrid setup, with lectures in seminar room 3418 at the math department at KTH, and as well on Zoom:
https://kth-se.zoom.us/j/69115122544 Links to an external site.
Meeting ID: 691 1512 2544
The notes from the Zoom lectures can be found under Files -> Zoom notes.
During the first lecture we will settle the schedule for the rest of the course so that it fits most students.
Goal: To understand and use basic methods and theory for numerical solution of partial differential equations.
Some topics: finite difference methods, finite element methods, multi grid methods, adaptive methods.
Some applications: elliptic problems (e.g. diffusion), parabolic problems (e.g. time-dependent diffusion), hyperbolic problems (e.g. convection), systems and nonlinear problems (conservation laws and optimal control).
Learning outcomes etc.
Prerequisites: undergraduate differential equations and numerics.
Literature:
- Stig Larsson and Vidar Thomee, Partial Differential Equations with Numerical Methods, Springer-Verlag (2009), ISBN 978-3--540-8870 5-8, (LT)
- Claes Johnson, Numerical Solution of Partial Differential Equations by the Finite Element Method, Dover Publication (2009), Cambri dge University Press (1988) (CJ)
- Adaptive FEM lecture notes (LN1)
- Finite difference methods lecture notes (LN 2)
- Optimal Control and Hamilton-Jacobi equations Download Optimal Control and Hamilton-Jacobi equations, lecture notes (LN 3)
The literature overlaps, so the list gives alternatives.
Plan:
- Introduction and overview: 1D elliptic finite element and finite difference methods, (LT chapters 2, 4.1, 5.1; CJ chapter 1, LN1)
- Elliptic equations, minimization and variational methods, maximum principle (LT 3,4,5; CJ 2, 3)
- A priori and a posteriori error estimates, adaptive methods (LT 5.4, 5.5; CJ 4; LN1)
- Solution methods, multigrid (LT B, CJ 6,7)
- Parabolic equations, Lax equivalence theorem, energy methods, finite difference and finite element methods (LT 8, 9,10; CJ 8; LN2)
- Hyperbolic problems, convection diffusion, artificial diffusion, streamline diffusion finite elements, conservation laws (LT 11,12 ,13; CJ 9)
- Hamilton-Jacobi equations and optimal control (LN 3)
Homeworks:
- Homework 1: Problem 1.2 (derive the wave equation), 2.1 (illustrate the maximum principle) in (LT) due February 8th.
- Homework 2: Problem 3.1 and 3.3 in (LT). Due February 21.
- Homework 3 Download Homework 3. Due March 21.
- Homework 4: Problem 3.5, 3.9 and 3.13 in (LT), due april 6.
- Homework 5: (4.5 or 4.4), (5.17 or 5.18) in (LT), and 6.7 in (LN2) Lecture notes on von Neumann stability, due May 11.
Final written exam
The final written exam takes place June 1st from 9-12 in lecture room V01. The questions in the exam will be a subset of the questions in this list Download this list.
Course inquiry
Please give your feedback on the course in the course inquiry.
Welcome!,
Mattias Sandberg