FSF3562 Numerical Methods for Partial Differential Equations 2020

Graduate course, starting January 15th 2020 at 15.15-17.00 in room F11 Department of Mathematics KTH, period 3-4, 7.5 ECTS

Schedule: Lectures once a week in period 3-4: from January 28 to March 10 on Tuesdays 10-12 in room F11, KTH, except Tuesday February 11th moved to Thursday February 13th at 10-12 in room F11. The lecture March 20 at 8-10 in F11, April 3-24 will be Fridays at 8-10 in room F11, and April 30 in room F11, May 7 at 8-10 in room 3418.

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)

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; LN3)

News:

2020-03-17: The remaining lectures will be online. There are recordings in the "Media Gallery" for lecture 10, which includes we skipped before: nonlinear multigrid, elliptic regularity, L^2 error estimates, blow-up of unstable schemes. There is also a pdf with lecture notes.

2020-03_20: Here is a pdf with comments on Computerlab 1. I put the list of questions for the take-home exam here.

2020-04-23 The Zoom-presentations of the projects are suggested to be May 26 starting at 9.00. Each presentation is 20 minutes. Put your slides in the related Canvas "Assignment".

Homework and computer lab:

Homework 1: Problem 1.2 (derive the wave equation), 2.1 (illustrate the maximum principle) in (LT) due February 11th.
Homework 2: do Problem 3.1 and 3.3 in (LT) (formulate variational form and prove existence of unique solution) , or do Computer lab 0, due February 26nd.
Computer lab 1 due March 17th.
Homework 3: Problem 3.5, 3.9 and 3.13 in LT, due 7/5
computer lab 2 due April 20th.
Homework 4: (4.5 or 4.4), (5.17 or 5.18) in (LT), and 6.7 in (LN2) Lecture notes on von Neumann stability, due 21/5.

Here is a list of questions for the take-home exam: hand in answers to the questions written with your own words, before June 5th.

Presentations (group of 1-2 presents a paper or chapter on numerics for PDE) for instance:

A. Brandt, Multi-level adaptive solutions of boundary value problems, Math. Comp. 31 (1977), 333-390.
Boundary element methods (CJ 10; LT 14.4)
Mixed finite elements ( CJ 11; LT 5.7)
Spectral methods (LT 14.2)
Stochastic elliptic PDE
Maxwell
MHD
Obstacle problems and variational inequalities
...

To pass the course requires at least 16 credits and passed computer laborations. A good solution of the whole take-home exam gives 10 credits. A good solution of a homework problem gives 1 credit. A good presentation gives 5 credits. Therefore one obtains 25 = 10 + 10x1 + 5 credits if everything is good.

Here is a link to a course enquiry for your comments to help improve the course (code: equation)

Welcome!
Anders Szepessy,
szepessy@kth.se, 790 7494

FSF3562 VT20-1 Numerical Methods for Partial Differential Equations