Math 676: Finite Element Methods in Scientific Computing

Spring 2021 — Section 600

MWF 1:35pm - 2:25pm in Blocker 113

Lecturer: Matthias Maier

Catalogue Description:

Basic finite element methods; structure of finite element codes; assembling linear systems of equations and algorithmic aspects; linear iterative solvers; adaptive mesh refinement; vector-valued and mixed problems; nonlinear problems; visualization; parallelization aspects. Additional topics may be chosen by instructor.

Target audience:

The course is designed for students involved in research in numerical methods, or students who want to use the finite element method for simulations in their graduate research.

Course overview and learning outcomes:

This course is intended to highlight some practical aspects of the finite element method, and more generally scientific software development. We will discuss in particular how typical finite element software is structured, how algorithmic details are efficiently implemented, and how necessary pre- and postprocessing steps are executed in the scientific computing workflow. The course will use the deal.II library (website), a large open source finite element library that has been developed over the last 20 years by an international team of researchers.

My goal for this course is that you (a) develop a solid understanding of modern scientific software development and the tools used for this, and (b) to develop a program based on deal.II that helps you in your graduate research. In particular, we will cover the following topics:

• Software development: The Unix eco system and toolchain, some advanced topics in C++, IDEs (Eclipse), distributed and collaborative version control (git), documentation (doxygen), software engineering aspects of large software projects,
• Finite element software: Basic finite element methods, structure of finite element codes, iterative solvers, preconditioners, nonlinear problems, time-dependent problems, hyperbolic problems
• deal.II: Installation and setup, assembling linear systems, algorithmic aspects, adaptive mesh refinement, software design and data structures. Additional topics based on interest.

Course format:

• Lectures: A significant portion of the lecture time is set aside to work with you on your individual research project. In order to facilitate this, I will assign external resources (documentation and video lectures) that you are required to read, or watch outside of class. We will discuss them briefly at the beginning of class.

• Projects: At the beginning of the course you will be assigned a project related to your research or interests. A large part of the class time is set aside to work on these projects, and a large portion of the final grade will be based on the project.

  Please visit the tutorial overview and in particular have a look at step-21, step-24, and step-25 for some project examples. If you are interested, good enough projects will be published as part of the library and distributed with future versions. Of course, you will then also be credited for your work.