SF3562 Numerical Methods for Partial Differential Equations 2017

Graduate course, starting September 5 (2017) at 10.15-12.00 in room 3418 Department of Mathematics KTH, period 1-2, 7.5 ECTS

Preliminary Schedule: Room 3418 Tuesdays 10-12; except Nov 3, Nov 16, Nov 20 also 10-12 in 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--540-88705-8, (ST)
Claes Johnson, Numerical Solution of Partial Differential Equations by the Finite Element Method, Dover Publication (2009), Cambridge University Press (1988) (CJ)
Adaptive FEM lecture notes (LN1)
Finite difference methods lecture notes (LN2)

The literature overlaps, so the list gives alternatives.

Plan:

Introduction and overview: 1D elliptic finite element and finite difference methods, (ST chapters 2, 4.1, 5.1; CJ chapter 1, LN1)
Elliptic equations, minimization and variational methods, maximum principle (ST 3,4,5; CJ 2, 3)
A priori and a posteriori error estimates, adaptive methods (ST 5.4, 5.5; CJ 4; LN1)
Solution methods, multigrid (ST B, CJ 6,7)
Parabolic equations, Lax equivalence theorem, energy methods, finite difference and finite element methods (ST 8, 9,10; CJ 8; LN2)
Hyperbolic problems, convection diffusion, artificial diffusion, streamline diffusion finite elements, conservation laws (ST 11,12,13; CJ 9; LN3)

News:

5/9: The course started 10.15-12 in room 3418 with examples of PDEs and some general numerical issues, including sections 1 and 4.1 in (LT). Next time we will do FD and FEM LT-chapters 2, 3.1-2, 4.1-2, 5.1-2.

12/9: We proved an error estimate for 1D FD by maximum principle in chapter 4.1 and presented the variational form and FEM for 1D chapter 3.5, 5.1. Next time comes the Lax-Milgram theorem and function spaces in chapter A1 together with the Neumann problem.

19/9: We formulated and proved LAx-Milgrams theorem in the case of symmetric bilinear forms, introduced weak derivatives (LT chapter A1-2), mixed Dirichlet Neumann boundary value problems (chapter 3.6). Next time we will study apriori and aposteriori error estimates for FEM (LT 5.4-6), regularity (LT 3.7), assymbly and quadature.

26/9: We derived a priori and a posteriori error estimates for FEM, studied regularity, assembly and quadrature. Next time we will discuss adaptive FEM in the lecture notes LN1 and Lax equivalence theorem. The adaptive FEM will be used in Homework 2 below.

3/10: We formulated adaptive FEM (CJ 4.6, LN1) and started studying Lax-equivalence theorem. Next time we continue with the equivalence theorem and von Neumann stability analysis (LN2 and chapter 9.1, 12.1-2) and solution methods (LT chapter 8, CJ 6,7). We will have no lecture October 17th.

30/10 Last lecture was on iterative methods and the multigrid method. Week 44 the lecture will be on Friday 10.15-12 on room 3418 and it will be about numerical methods for parabolic and hyperbolic problems.

3/11 The lecture was on numerical methods for parabolic problems and on properties of convection and convection-diffusion equations. Next time we will study numerical methods for wave equations and some nonlinear PDE.

Homework and computer lab:

Problem 1.2 (derive the wave equation), 2.1 (illustrate the maximum principle) in (LT) due 19/9.
Problem 3.1 and 3.3 in (LT) (formulate variational form and prove existence of unique solution) due 3/10.
Homework 2 due week 44.
Problem 3.5, 3.9 and 3.13 in LT, due 24/10
computer lab 2 due week 51.
(4.5 or 4.4), (5.17 or 5.18) in (LT), and 6.7 in (LN2) Lecture notes on von Neumann stability, due december 12th.

Here is a preliminary list of questions to prepare for the exam. The exam consists of five questions related to the list.

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

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

To pass the course requires at least 16 credits. A good answer on a question in the exam gives 2 credits. A good solution of a homework problem gives 1 credit. A good presentation gives 5 credits. Therefore one obtains 25 = 5x2 + 10x1 + 5 credits if everything is good.

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