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 content

• 16 lectures
• 8 laboratory sessions

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 that are based in large parts on the following books. 

[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

Lab 1: Krylov methods

Lab 2: ODE time stepping

Lab 3: FEM assembly

Lab 4: PDE and FEM in 1D/2D

Lab 5: Adaptive FEM

Lab 6: Optimization

Week plan

Week 1 

• Lecture 1: Vector spaces
• Lecture 2: Linear transformations
• Lecture 3: Linear systems of equations - direct methods

Week 2 

• Lecture 4: Eigenvalue problems 
• Lecture 5: Iterative methods - Krylov methods 
• Lab 1: Krylov methods

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