- BO = Note on finite difference methods by Brynjulf Owren
- CC = Note on finite element methods by Charles Curry
The following schedule is very tentative
|Week||Subjects||Reading||Some relevant (exam) questions|
|2|| Introduction to the course |
| BO section 2, 3-3.1 |
Introduction to the course. Difference formulas and difference operators. Numerical solution of the 1-dimensional heat equation.
|3||Boundary value problems|| BO section 3. |
Error analysis of a difference formula of the 1-dimensional Poisson problem.
Truncation error, consistency, convergence.
Dirichlet, Neumann and mixed boundary conditions
|June 2014, Problem 5.|
|4||Parabolic equations|| BO section 4. |
Forward Euler, Backward Euler and Crank-Nicolson.
Stability analysis for the heat equation.
Semidiscretization of linear and nonlinear PDEs.
Boundary conditions were not discussed in the lectures, but the idea is the same as for BVP, so it is part of the curriculum.
|5||Stability, consistency and convergence|| BO section 5, except 5.5 |
The concepts stability, consistency, convergence. Sufficient conditions for convergence, and Lax Equivalence theorem. You should also be able to do simple convergence proofs. Von Neumann stability analysis.
|June 2018, Problem 2. August 2013, Problem 2. June 2013, Problem 3 and 4. May 2011, Problem 2. June 2010, Problem 3.|
|6|| Elliptic equations |
Model: The Poisson equation
| BO section 6.1-6.2 and 6.9-10. |
The 5-point formula for the Poisson equation. The discrete maximum principle and a convergence of the 5-point formula.
|August 2014, problem 2. June 2014, problem 2. June 2010, problem 1.|
|7|| BO section 6.3-6.5. Discretisation on boundaries. |
|8|| Hyperbolic problems |
Model: The wave equation and the transport equation.
| BO section 6.6-6.9. Discretisation on irregular grid. Box-integration. |
BO 7.1 and 7.2 Hyperbolic equations.
Examples. Characteristics. Conservation laws.
|June 2018, problem 1. May 2016, problem 2. May 2015, problem 1. August 2014, problem 4. June 2014, problem 4. June 2012, problem 3.|
|9|| Project release |
some time during the week.
|BO section 7.3 - 7.7|
|10||Finite element methods|| CC section 1, 2.1-2.2, 3.1, 5.1, 5.2 |
Setting up the variational problem, Galerkins method, mesh, and basis functions for FEM in 1 and 2 dimensions.
|June 2018, problem 3. May 2016, problem 1. May 2015, problem 4.|
|11|| CC section 2.3-2.6, |
Boundary conditions and the assembly process in 1D.
|12||CC section 3 and 4: Existence and uniqueness of the variational form. Convergence.||May 2017, problem 1.|
|13||CC section 5.3-5: More on 2D-problems.||June 2012, problem 4.|
|15||Numerical linear algebra|| Classical iterative techniques: Jacobi, Gauss-Seidel, SOR. |
Strikwerda, chapter 13.1-3 or Quarteroni et.al. chapter 4.2.1-2.
Line search methods: Steepest descent and the conjugate gradient method.
Strikwerda chapter 14.1-3 or Quarteroni et.al. chapter 4.3.3-4.
Focus on the idea of the methods, convergence criterias (no proofs) and how they can be implemented for discretized PDEs.
|May 2017, problem 3. June 2014, problem 3. May 2013, problem 2. August 2013, problem 2c).|
Here are the theoretical course prerequisites. If some of this seems unfamiliar, new, or forgotten, do not worry too much: relevant parts will be revised during the semester, and the course note covers many topics. The first two rows of the table below are the most important and priority should be given here.
|Calculus||Taylor's theorem. Existence and uniquenes of solutions of ODEs. Solution of simple ODEs and simple (linear) PDEs. Fourier series and transform. Norms and function spaces.|
|Linear algebra||Basics||Vector and matrix norms; Symmetric Positive Definite matrices; Inner product spaces; Linear independence; basis of a vector space; orthonormal basis.|
|Matrix factorizations||Diagonalization and orthogonal diagonalization of matrices; LU decomposition (Gaussian elimination, pivoting); Cholesky factorization; QR factorization; SVD; Jordan Canonical form; Schur factorization.|
|Topics of interest in numerical linear algebra||Spectral radius; Gershgorin's theorem; Condition number; Neumann series.|
|Iterative methods||Newton method; fixed point iteration; convergence of the basic iterative methods (Jacobi, Gauss-Seidel and SOR) for linear systems.|
|Numerical ODEs||Runge-Kutta and multistep-methods, convergence of the Euler method, order conditions and stability, see the note from TMA4215 Numerical mathematics.|