$$ \newcommand{\uex}{u_{\mbox{\footnotesize e}}} \newcommand{\uexd}[1]{u_{\mbox{\footnotesize e}, #1}} \newcommand{\vex}{v_{\mbox{\footnotesize e}}} \newcommand{\vexd}[1]{v_{\mbox{\footnotesize e}, #1}} \newcommand{\Aex}{A_{\mbox{\footnotesize e}}} \newcommand{\half}{\frac{1}{2}} \newcommand{\halfi}{{1/2}} \newcommand{\tp}{\thinspace .} \newcommand{\Ddt}[1]{\frac{D #1}{dt}} \newcommand{\E}[1]{\hbox{E}\lbrack #1 \rbrack} \newcommand{\Var}[1]{\hbox{Var}\lbrack #1 \rbrack} \newcommand{\Std}[1]{\hbox{Std}\lbrack #1 \rbrack} \newcommand{\xpoint}{\boldsymbol{x}} \newcommand{\normalvec}{\boldsymbol{n}} \newcommand{\Oof}[1]{\mathcal{O}(#1)} \newcommand{\x}{\boldsymbol{x}} \newcommand{\X}{\boldsymbol{X}} \renewcommand{\u}{\boldsymbol{u}} \renewcommand{\v}{\boldsymbol{v}} \newcommand{\w}{\boldsymbol{w}} \newcommand{\V}{\boldsymbol{V}} \newcommand{\e}{\boldsymbol{e}} \newcommand{\f}{\boldsymbol{f}} \newcommand{\F}{\boldsymbol{F}} \newcommand{\stress}{\boldsymbol{\sigma}} \newcommand{\strain}{\boldsymbol{\varepsilon}} \newcommand{\stressc}{{\sigma}} \newcommand{\strainc}{{\varepsilon}} \newcommand{\I}{\boldsymbol{I}} \newcommand{\T}{\boldsymbol{T}} \newcommand{\dfc}{\alpha} % diffusion coefficient \newcommand{\ii}{\boldsymbol{i}} \newcommand{\jj}{\boldsymbol{j}} \newcommand{\kk}{\boldsymbol{k}} \newcommand{\ir}{\boldsymbol{i}_r} \newcommand{\ith}{\boldsymbol{i}_{\theta}} \newcommand{\iz}{\boldsymbol{i}_z} \newcommand{\Ix}{\mathcal{I}_x} \newcommand{\Iy}{\mathcal{I}_y} \newcommand{\Iz}{\mathcal{I}_z} \newcommand{\It}{\mathcal{I}_t} \newcommand{\If}{\mathcal{I}_s} % for FEM \newcommand{\Ifd}{{I_d}} % for FEM \newcommand{\Ifb}{{I_b}} % for FEM \newcommand{\setb}[1]{#1^0} % set begin \newcommand{\sete}[1]{#1^{-1}} % set end \newcommand{\setl}[1]{#1^-} \newcommand{\setr}[1]{#1^+} \newcommand{\seti}[1]{#1^i} \newcommand{\sequencei}[1]{\left\{ {#1}_i \right\}_{i\in\If}} \newcommand{\basphi}{\varphi} \newcommand{\baspsi}{\psi} \newcommand{\refphi}{\tilde\basphi} \newcommand{\psib}{\boldsymbol{\psi}} \newcommand{\sinL}[1]{\sin\left((#1+1)\pi\frac{x}{L}\right)} \newcommand{\xno}[1]{x_{#1}} \newcommand{\Xno}[1]{X_{(#1)}} \newcommand{\yno}[1]{y_{#1}} \newcommand{\Yno}[1]{Y_{(#1)}} \newcommand{\xdno}[1]{\boldsymbol{x}_{#1}} \newcommand{\dX}{\, \mathrm{d}X} \newcommand{\dx}{\, \mathrm{d}x} \newcommand{\ds}{\, \mathrm{d}s} \newcommand{\Real}{\mathbb{R}} \newcommand{\Integerp}{\mathbb{N}} \newcommand{\Integer}{\mathbb{Z}} $$ Course ambitions and future improvements

Course ambitions and future improvements

HPL

Dec 14, 2013

Long-term goals

The goal of INF5620 is to provide a series of modules with all the fundamental building blocks for solving PDE problems by finite difference, finite element, and finite volume methods, but with particular emphasis on finite element methods.
A subset of the modules will constitute the INF5620 course.
A hope is to migrate the first modules on finite difference methods to the 2nd semester in the MIT and FAM bachelor programs at UiO.
Other bachelor courses, especially in physics and mechanics, can hopefully make use of some modules as background material for PDE-based projects.
INF5620 can easily adapt the composition of modules to a changing environment (see below for some challenges connected to this).
The philosophy of each module is to

present a gentle and intuitive introduction to the subject,
use the smplify-understand-generalize approach,
use as simple mathematics as possible,
provide all nuts and bolts necessary to get the numerics to play in software,
introduce appropriate notation such that the step from mathematics to software is as small as possible,
explore the quality of numerics through exact solutions of the discrete equations (such as Fourier wave components) and reach more detailed insight than what is possible with traditional asymtotic convergence theories (such truncation error analysis),
emphasize program verification through MMS and other methods,
understand the how PDEs arise and are applied to study physical problems.

It is a clear ambition to publish most modules in one or more books, yet with HTML-based versions for various electronic devices.
Another ambition is to make the modules open and accessible for other universities and self study.
The focus will remain on the introductory and fundamental material, ideally in a form accessible with a minimum of mathematics and programming at the early bachelor level. There is a lot of advanced literature available once the fundamentals are understood.
The students come to the course with a great variety in background, and we have effective means to deal with this heterogeneous audience.

Ultimate goal for UiO. Students from UiO know how to solve all the PDEs they have seen.

Planned improvements

Bitbucket/GitHub

Communicating through a version-controlled repo is in theory a good idea because it easily handles a lot of files. (This is also the way professionals work, and master students should use version control systems for their thesis.)

Experience 2013:

With many students ($ >10 $) the teachers need a much more streamlined setup with much less freedom and no tedious and error-prone clicking in web interfaces.
Due to a technical misunderstanding and then a shortcoming of the Bitbucket web interface, the feedback on oblig2 was delayed by more than a week. The semi-automatic system does not scale to a class of 40+ students.

Suggestions:

Choose GitHub or Bitbucket, probably the latter. Require Git as version control system.
Make streamlined setup with strict naming conventions for directories, files, student information (the full name for instance...), etc.
Provide a simple step-by-step recipe for the student in INF5620.
Develop more scripts to automated forking, fetching upstream, pushing to fork and pull requests. (Need to explore the Bitbucket API to avoid manual interaction with web pages, but it seems possible).
Need some kind of notification when a student is finished with a project (now we have to fetch upstream from the student account and look for something that can be the name of a new project...).

Remark. Devilry plans to develop a model that is very similar to the setup with Bitbucket/GitHub in INF5620, and Devilry might then be a more user-friendly and automated platform than Bitbucket.

Compulsory exercises

Experience 2013:

With 40-50 students the work with feedback on compulsory exercises is too comprehensive and takes too much time. (Scaling up with more teaching assistants is probably not an option in a 5xxx course.)

Suggestions:

Keep oblig1 as is to ensure familiarity with basic finite differences, programming and (in particular) testing.
Reduce the amount of work in oblig2 and define the rest and more as an INF5631 project (5 p).
Let two and two students give feedback on two projects with the teacher available in the room. This can happen the week after the deadline. (Technical note: students can fork the teacher's fork of the students private repo.)
Use this setup for FEM for the wave equation and the default version of the final project.
The teacher gives feedback to students who define their own final project.

Lectures

Setup 2013:

Lectures have switched between slides and handwritings.
Handwritings are usually detailed and ensure a slow progression.
Slides allow flexibility: quick overview or more detailed treatment, with (much) less errors.
Slides as reading material (study guides) are meant as a first overview and as a repetition of the key points.

Experience 2013:

4h per week are needed for lectures - that's quite much and it's all a one-way communication.
Most of theory is example-driven, yet it would be good to solve more plain exercises. And more important: there should be time for more individual guidance.

Suggestions:

Short-term alternative: more of the topics are treated as structured self-study with a clear plan how on how the students should work.
More long-term alternative: all the lectures are available in video format and the face-to-face teaching is primarily used for detailed explanations, problem solving, and supervision in project.
Also: videos could be in two versions, one quick overview and one detailed treatment.
2 h lectures on very quick theory overview, students-driven questions, and solving exercises.
2 h lab where the teacher is available for individual quidance.

Preliminary experience with self-study of theory (video or reading).

The lectures become more demanding as we dive deeper into details in theory or problems and develop a two-way communication.
The lectures require preparation, with a likely danger that the lectures are meaningful only for a fraction of the students.

Recall: most lectures in the traditional one-way communication format are meaningful and quite easy to follow without prepratations.

Thoughts about the 2014 version

Goal (probably realistic). Have all written material available at the beginning of the course.

The 2013 progression of topics is very natural - in a table of contents - but places the most difficult topics at the end. More time for digesting finite elements could be beneficial.

Start with a quick intro/overview of finite differences for ODEs, wave and heat equations
Start early with the more theoretical topics:

Approximation of functions
Galerkin, least squares etc for simple differential equations
The finite element machinery (at least the simpler parts)

Continue with finite difference topics:

Structured self-study for those with weak background in finite difference methods
Focus on implementation and testing
Handling of various types of boundary conditions, variable coefficients
A smaller version of "Oblig2" on 2D wave equations
Numerical artifacts and their analysis

Diffusion equations: finite elements and finite differences
Nonlinear PDEs
Systems of PDEs:

The Navier-Stokes equations
The equations of elasticity
Coupled flow-heat transport