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

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.Länkar till en externa sida.

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 FEMLänkar till en externa sida. lecture notes (LN1)
Finite difference methodsLänkar till en externa sida. lecture notes (LN 2)
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. 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.

Course inquiry

Please give your feedback on the course in the course inquiry.

Welcome!,
Mattias Sandberg

FSF3562 VT22 Numerical Methods for Partial Differential Equations (60860)