# Compatible Finite Elements for Problems in Mixed Form

In this lecture we will analyze several model problems of physical importance (such as Poisson, Stokes and Maxwell equations), and their approximation by mixed finite element methods. By expressing them in the framework of Hilbert complexes we will better understand some guiding principles of compatible finite element methods which preserve their structure at the discrete level. In particular, we will see how some key stability and accuracy properties of compatible finite element methods derive from this structure. By reviewing some important examples based on spline and spectral finite element spaces, students will be able to derive by themselves the building blocks of such compatible finite element methods. Compatible finite element methods (FEM) are frequently used in the context of plasma physics simulations.**Until further notice, the lecture will be given online (details below).**

## News

- The first lecture will be on Friday, April 16, 2021.
- Lectures will be held on Fridays, 10:15-11:45 in this BigBlueButton classroom.
- The link and access code to the virtual classroom will be sent to all registered participants, and may also be given on demand (write me at martin.campos-pinto[at]ipp.mpg.de).

## Literature

- D.N. Arnold, Finite Element Exterior Calculus, SIAM (2018)
- D.N. Arnold, R.S. Falk and R. Winther, Finite element exterior calculus, homological techniques, and applications, Acta Numerica, (2006)
- D.N. Arnold, R.S. Falk and R. Winther, Finite element exterior calculus: from Hodge theory to numerical stability, Bull. Amer. Math. Soc. 47, (2010)
- D. Boffi, F. Brezzi and M. Fortin, Mixed finite element methods and applications, Springer Series in Computational Mathematics 44 (2013)
- H. Brezis, Functional analysis, Sobolev spaces and partial differential equations, Springer (2010)
- R. Hiptmair. Maxwell’s Equations: Continuous and Discrete. Computational Electromagnetism, Lecture Notes in Mathematics 2148, Springer (2015)
- F. Assous, P. Ciarlet and S. Labrunie, Mathematical foundations of computational electromagnetism, Springer (2018)

## Lecturer

- Dr. Martin Campos-Pinto.

## Finalized lecture notes

- tum_lecture_notes_2021_complete.pdf: pdf document

## Notes for Class 12' of July 11

- tum_class_notes_2021_july_11.pdf: online notes

## Notes for Class 12 of July 9

- tum_class_notes_2021_july_9.pdf: online notes

## Notes for Class 11 of July 6

- tum_class_notes_2021_july_6.pdf: online notes

## Notes for Class 9-10 of June 24-25

- tum_class_notes_2021_june_24_25.pdf: online notes
- tum_lecture_notes_2021_class_june_24.pdf: incomplete notes

## Notes for Class 8 of June 11

- tum_class_notes_2021_june_11.pdf: online notes
- tum_lecture_notes_2021_class_June_11.pdf: incomplete notes

## Notes for Class 7 of May 28

- tum_class_notes_2021_may_28.pdf: online notes
- tum_lecture_notes_2021_class_may_28.pdf: incomplete notes

## Notes for Class 6 of May 21

- tum_class_notes_2021_May_21.pdf: online notes
- tum_lecture_notes_2021_class_may_21.pdf: incomplete notes

## Notes for Class 5 of May 17

- tum_class_notes_2021_May_17.pdf: notes from online Class 5 (anwsers to some questions)

## Notes for Class 4 of May 7

- tum_class_notes_2021_May_7.pdf: online notes
- tum_lecture_notes_2021_May_7.pdf: incomplete notes

## Lecture notes for Classes 1-3

- tum_lecture_notes_2021_April_30.pdf: lecture notes from Class 3
- tum_lecture_notes_2021_April_23.pdf: lecture notes from Class 2
- tum_lecture_notes_2021_April_16.pdf: lecture notes from Class 1

- drt041.pdf: drt041.pdf

- LPG_jcp_2018.pdf: LPG_jcp_2018.pdf