Functional Analysis Essentials for the Finite Element Method (FAEFEM)

This course was held in spring 2023.

Course Description:

This course focuses on the essentials of functional analysis that are directly useful in designing, understanding, and analyzing finite element methods. Such methods present an efficient tool in approximating solutions to partial differential equations (PDEs). With the availability of platforms like FEniCS, such methods have become increasingly accessible. The course adopts a functional analytic perspective to explore why, how, and when such methods work well or fail. Completing this course provides a broader access to the mathematical literature which enables participants to transfer ideas to their relevant applications.

Learning Outcomes:

After completing this course, you will be able to:

1. Identify the correct functional setting for weak formulations of PDEs and of their respective discretizations. 
2. Apply results from functional analysis to show that solutions to different discrete approximations (primary, mixed, and saddle point formulations) exist and are unique. 
3. Analyze the approximation error of finite element methods using tools from functional analysis.

Target group:

This course is intended for researchers and students who use finite element methods for their simulations, and want to improve their knowledge of the theoretical aspects of such methods.

• Postdocs, PhD students, and Master's students at Simula, as well as partners from the Univeristy of Oslo, that are working in scientific computing or applied mathematics. A background (undergraduate level is sufficient) in real analysis and linear algebra is required.
• Senior undergraduate students in mathematics or applied mathematics who have completed classes in real analysis and in linear algebra.

Program:

This class will be held over 3 weeks in March 2023 at Simula Research Laboratory (Kristian Augusts gate 23). In order to make this class as relevant as possible to your research, there is one short assignment per week to reinforce your learning.

Week 1
Class/lecture: Friday, 17 March, 10:00-13:00 in Nordmarka
Assignment: participants are asked to submit a 1 page report on one problem (PDE) that you have encountered in your research or studies, by the end of week 1.

Week 2
Class/lecture: Friday, 24 March, 10:00-13:00 in Nordmarka
Assignment: participants are asked to submit a more precise problem statement, including function spaces and weak formulation, for the problem submitted as a short 1 page report in week 1.

Week 3
Friday, 31 March, 10:00-14:00* in Nordmarka
Assignment: participants will hold short presentations (10 min each) or submit a report (1-5 pages) on an application(s) of the discussed theorem(s) in the class addressing aspect(s) of their problem.

*An additional 1 hour session will be held on the last day of class. This last session will consist of short student presentations.

Resources:

• Lecture notes on the material discussed in the course will be provided.
• Office hours will be available to participants, during which the instructors will be availble for 1-2 hours a week at Simula to discuss any questions you have on the course material or the student presentations at the end of the course.
• Contact:
  For questions related to the course set up and material, please contact Marius Zeinhofer (mariusz@simula.no) or Rami Masri (rami@simula.no)
  For practical questions, please contact Simula Academy (academy@simula.no)

Admission:

As the course capacity is limited, we may not be able to accommodate all applicants. In order to prioritize admission for those that this course is most relevant for, please complete the application form below.

Note: There are additional conditional questions based on whether you are a student, PhD candidate or postdoctoral fellow.

Application is now closed. All applicants will get feedback on their application by Feb 25.