Homogenization II: Stochastic problems

This is a supplementary/advanced course for the Maxwell Institute Graduate School in Analysis and its Applications (MIGSAA).
Detailed information about the lectures are available on the Scottish Mathematical Sciences Training Center (SMSTC) website.

Numerical Methods for PDEs

Detailed information about the lecture is available on VISION. The Jupyter iPython notes of the first introductory lecture can be access here: Lecture1_Introduction.ipynb .

The following beamer slides represent the content of each lecture:

Week 1
Parabolic equations
Lecture 1:
Introduction, Classification of PDEs
Lecture 2:
Diffusion/Heat conduction, Finite Differences
Lecture 3:
FTCS scheme/Examples
Week 2
Parabolic equations
Lecture 4:
Local truncaton error/Matrix form of FTCS scheme
Lecture 5:
Stability: Eigenvalue & von Neumann method
Lecture 6:
BTCS scheme/The θ-method & its LTE
Week 3
Parabolic equations
Lecture 7:
Stability & matrix form of the θ-method
Lecture 8:
More general PDEs & boundary conditions
Lecture 9:
Multilevel schemes/Convergence/Lax equivalence/Examples
Week 4
Parabolic equations
Lecture 10:
FTCS scheme in 2D/2D θ-method
Lecture 11:
The ADI method
Lecture 12:
Stability of the ADI method/LTE & stability of a nonlinear example
Week 5
Hyperbolic equations
Lecture 13:
Introduction to Hyperbolic PDEs/The advection equation
Lecture 14:
LTE & stability of the FTBS and FTFS schemes/The FTCS scheme
Lecture 15:
Relative phase error/Test problems/The leapfrog scheme
Week 6
Hyperbolic equations
Lecture 16:
The leapfrog scheme (LTE, stability & phase error) and the Lax-Wendroff scheme (LTE, stability & phase error)
Lecture 17:
Backward time schemes/Crank-Nicolson scheme (LTE, stability & phase error)/Wave equation (LTE, stability & phase error)
Lecture 18:
Coupled system/Nonlinear conservation laws/A nonlinear Lax-Wendroff scheme
Week 7
Elliptic equations
Lecture 19:
Central difference approximation/Local truncation error/Convergence
Lecture 20:
Variational formulation/Linear FEM 1D
Lecture 21:
Linear FEM 2D
Week 8
Elliptic equations
Lecture 22
Linear FEM 2D (contin. of Lect. 21)
Lecture 23
FEM 2D/Sobolev spaces/Stability/Error estimates
Lecture 24
(Continuation of Lecture 23: Error estimates)
Week 9
FD/FEM examples
Lecture 25
Examples of elliptic problems: Conductivity (resistor networks)/Porous media flow
Lecture 26
Sample exam part 1
Lecture 27
Sample exam part 2
Week 10
FD/FEM examples
Lecture 28
Discussion of sample exam
Lecture 29
Discussion of sample exam
Lecture 30
Discussion of further sample problems

The slides above are based on the detailed lecture notes from previous years
which can be downloaded here (or from VISION): lectureNotes.pdf

Possible further reading:
B.S. Jovanović and E. Süli, Analysis of Finite Difference Schemes, Springer 2014.