Teaching

MATH 664-600: Computational Software for Large-Scale PDE Solvers

Course Outline

The course is intended to bring students to a level where they can be active users or, participants in and developers of projects that aim at the large-scale solution of partial differential equations by the finite element method, possibly using parallel and distributed computing. One of the most important lessons learned over the past decade is that due to their size such software can't be written from scratch for each new project, and must instead build on existing software libraries that handle most of what constitutes the finite element method as well as linear algebra and parallel communication. Applications then only have to implement things like bilinear forms, outer nonlinear solution loops, and linear solvers specialized to the application.

In the course, students will be taught how to build such software on top of existing libraries, in particular the deal.II adaptive finite element library as well as the PETSc parallel linear algebra library. The first part of the course will be used to review the basic mathematical concepts used in this software, such as finite element theory and iterative solvers. The rest of the course will be used for projects in which students are guided through the implementation of solvers for applications from their particular field. Part of this will be teaching the use of software engineering practices, such as the use of revision control management systems (e.g., CVS or Subversion), writing documentation, and writing tests for automated testsuites. Projects will be picked according to the field of research of a student or her interests. The goal is the development of a code that a) helps in the research of a student, and b) may be used as the starting point for future generations of students.

An outline of the topics to be covered is as follows:

• Basics of finite element implementations
• Basics of iterative solvers for large linear systems
• Basics of iterative nonlinear solvers
• Use of existing software libraries for partial differential equations and linear algebra
• Parallelization strategies for distributed computing and challenges to parallelization, such as adaptivity (depending whether there is demand for this topic)
• Collaborative software development, e.g. the use of CVS or subversion
• Software engineering practices for large-scale software, e.g. automated build and test systems, tools for documentation

This course is intended for students in the engineering disciplines as well as students of mathematics involved in research in numerics. Feedback from the involved groups would therefore be appreciated, and I will make efforts to incorporate suggestions on other topics that might be of importance to these groups.

Prerequisites

• Knowledge of the finite element method. This is covered by MATH 610, though we will probably need little of the theory developed there. An engineering equivalent of this course would definitely be sufficient. Good students of MATH 609 should also be able to read up quickly on how the finite element method works, though they may lack an understanding of why.
• Basics of iterative solvers. MATH 609 covers some of this material, and MATH 639 covers it in detail. Good students of MATH 609 should do fine, though.
• Good knowledge of C++. Students without adequate C++ programming skills will not be able to benefit from this course, and in contrast to content material, the experience needed to write programs can't be taught in a few classes at the beginning of the semester. All successful large-scale software packages are written in advanced programming languages, and students should have experience in object-oriented programming and preferrably the use of templates in C++.

Literature

To the best of my knowledge, there isn't much in terms of books on writing large-scale software for finite element simulations. What is there often treats a single software package; for example, there are books about FEMLAB and DiffPack, two commercial finite element packages. For the two packages to be used in class, there are no such books, but PETSc comes with a manual of several hundred pages, and deal.II has several thousand pages of documentation. They can be found at

• the PETSc homepage, and
• The deal.II homepage.

For the other topics, there are good books. There is a plethora of books about the theory of finite elements; for iterative solvers of linear and nonlinear systems, the following books are commonly referenced:

• Y. Saad, "Iterative Methods for Sparse Linear Systems" (an exhaustive book on the theory of iterative solvers)
• R. Barrett et al., "Templates for the Solution of Linear Systems" (a short book explaining the implementation of a variety of linear solvers)
• C. T. Kelley, "Iterative Methods for Linear and Nonlinear Equations"

The use of Subversion is explained in

• C. M. Pilato et al., "Version Control with Subversion" (this book is available online, just click on the title)

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