Methods in Scientific Computing (DD2365), 7.5hp, Spring 2017

Course goals

The goal of the course is to present general and efficient numerical methods and algorithms for basic models of computational science, in particular particle models, ordinary differential equations (ODE) and partial differential equations (PDE). Research challenges in the field are highlighted, e.g. with respect to parallel and distributed computing.

Course activites

• 16 lectures
• 8 laboratory sessions

Examination

• Written examination, 4.5hp (Grade: A, B, C, D, E, FX, F)
  • Written exam based on the lecture notes of the course, and the lab modules Lab1-Lab6. 
  • Exam maximum 50p: 40p (A), 35p (B), 30p (C), 25p (D), 20p (E), <20p (F)
• Lab modules Lab1-Lab6, 3.0hp (P/F)
  • Lab modules: each LabX consists of a set of mandatory problems  to complete (P/F) + a set of optional problems that give bonus points for the written exam: max 2p/LabX (max 12p in total).
  • Deadline for lab reports: March 14 (After this date no bonus points will be awarded.) 
  • Lab reports for each module should be submitted online. 

Teachers

Johan Hoffman
Office hours: Tuesdays 9:00-10:00 (Office 4432, Lindstedtsvägen 5)

Johan Jansson

Niclas Jansson

Tania Bakhos 

Van Dang Nguyen 

Literature

Lecture notes distributed in time for the lectures.  

Extra reading 

[LIN] Trefethen, Bau, "Numerical Linear Algebra", SIAM, (ISBN 978-0-898713-61-9) [SIAM]

[CDE] Eriksson, Estep, Hansbo, Johnson, "Computational Differential Equations", Studentlitteratur, (ISBN 91-44-49311-8), 1996. [Bokus] [Studentlitteratur] [Kårbokhandeln]

NOTE: In some electronic versions of the CDE book, the Chapter 13 "Calculus of Several Variables" may be missing. For the problem sets the correct chapters should be: Chapter 8 "Two-Point Boundary Value Problems", Chapter 15: "The Poisson Equation", Chapter 21: "The Power of Abstraction". 

Lab modules

Lab1: Iterative methods for solving linear systems

Lab2: ODE time stepping

Lab3: FEM assembly

Lab4: PDE and FEM in 1D/2D

Lab5: Adaptive FEM

Lab6: Optimization

Software 

SciPy - Python-based ecosystem of open-source software for mathematics, science, and engineering

FEniCS project - computing platform for partial differential equations (PDE) 

Week plan

Week 1 

• Lecture 1: Vector spaces
  [Lecture notes 1]
• Lecture 2: Matrix algebra
  [Lecture notes 2]
• Lecture 3: Linear systems of equations - direct methods

Week 2 

• Lecture 4: Eigenvalues and eigenvectors 
• Lecture 5: Iterative methods
• Lab 1: Iterative methods for solving linear systems

Week 3 

• Lecture 6: Nonlinear equations - Newton method
• Lecture 7: ODE - time stepping/quadrature in 1D
• Lab 2: ODE time stepping

Week 4 

• Lecture 8: ODE models
• Lecture 9: Function approximation - piecewise polynomials, interpolation, LS/L2-projection
• Lecture 10: Quadrature in 2D/3D - quadrature, mesh, reference element, assembly algorithm  

Week 5

• Lecture 11: PDE - FEM for 1D BVP model problems
• Lecture 12: PDE - FEM for 2D/3D BVP model problems
• Lecture 13: PDE - semi-discretization of PDE IVP

Week 6 

• Lecture 14: Optimization - Adaptive FEM
• Lecture 15: Optimization - PDE constrained optimization
• Lecture 16: Course review

Week 7 

• Lab 3: FEM Assembly
• Lab 4: PDE and FEM in 1D/2D 
• Lab 5: Adaptive FEM

Week 8

• Lab 6: Optimization
• Lab 7: Lab review
• Lab 8: Lab review 

Week 9

Written exam