Computational PDEs I (SS 21)

Lecture held by:	Dietmar Gallistl
Dates:	We meet on Fridays 12-14 for the lecture (BBB link) and on Tuesdays 12-14 for the problem class.
Problem sheets:	Please hand in your solutions or questions with Dr. Olkhovskiy by Monday 9:00.

Further information (in German)

Lecture and exercises: Dietmar Gallistl
Friedolin info
Dates: see table below
Required preliminaries: basic knowledge on the finite element method^*; basic programming skills; basic linear functional analysis (Höhere Analysis I can be attendend in parallel)
(^* for example from one of the courses Iterative solvers for PDEs / Computational PDEs I / Numerical Analysis of instationary PDEs)
If you meet these requirements, you can follow the course without having attended Computational PDEs I.
This is a 6 ECTS course. We will meet roughly every second week (see schedule below) 8-12 for lectures and exercises. It is essential for the students to solve the theoretical problems and the programming assignments.
Oral exams on individual appointment.

Saddle-point problems; mixed finite elements; Stokes equations; linear elasticity; a posteriori error estimation

There are lecture notes: pdf file (regularly updated). The file also contains sections with problems and programming exercises.
Last update: 7-July-2025
Some software for getting started: LINK

See the bibliography in the lecture notes.

Problem solving: The table below displays the exercises to be discussed in the respective week (e.g., the exercises formulated in week 19 can be found in the row for week 20).
Legend:
orange=lecture

magenta=programming project (no class, individual appointments)

grey=no class (holiday)

Week	Friday	Exercises	Syllabus	References
15	11 Apr		BNB lemma, saddle-point problems	§2.1.1; Lemma 2.9
16	18 Apr
17	25 Apr	Problems 2.1--2.7	Brezzi splitting, Stokes equations	§2.1.3--2.2.1
18	02 May
19	09 May	Problems 2.8--2.12; Problems 1.22--1.23; Problem 2.17	Galerkin method, quasi-interpolation	§2.1.2--2.1.3, §1.1.3
20	16 May	Problems 1.35, 1.36, 1.40, 1.41, 1.44, 1.47	quasi-interpolation, Mini element	§1.1.3, §2.2.2--2.2.3
21	23 May
22	30 May
23	06 Jun	Problems 1.43, 1.46, 1.48, 1.49; Problems 2.13--2.15; Problem 2.18	duality, mixed Laplacian, H(div) and its traces	§2.3.1--2.3.2
24	13 Jun	Problems 2.19--2.25	Raviart-Thomas element	§2.3.3
25	20 Jun	Problems 2.25--2.27, 2.31--2.34	details on traces and H^1/2 spaces	§3.1
26	27 Jun	Problem 2.28
27	04 Jul	Problem 2.28, Problems 3.1--3.6, Lemma 3.11	linear elasticity, locking	§4.1
28	11 Jul	Problems 4.1--4.4	hypercircle, a posteriori error estimates	§2.3.6

[GT98] Gilbarg/Trudinger. Elliptic PDE of second order. Springer, 1998.
[MPS00] Maugeri/Palagachev/Softova. Elliptic and parabolic equations with discontinuous coefficients. Wiley, 2000.
[SS13] I. Smears, E. Süli. Discontinuous Galerkin finite element approximation of nondivergence form elliptic equations with Cordes coefficients, SIAM Journal on Numerical Analysis 51(4), pages 2088--2106, 2013. doi:10.1137/120899613 download
[SS14] I. Smears, E. Süli. Discontinuous Galerkin finite element approximation of Hamilton–Jacobi–Bellman equations with Cordes coefficients, SIAM Journal on Numerical Analysis 52(2), pages 993--1016, 2014. doi:10.1137/130909536 download
[T25] Finite element approximation for uniformly elliptic linear PDE of second order in nondivergence form, Math. Comp. 94, pages 1043--1064. 2025. doi:10.1090/mcom/3966 download
[G18] D. Gallistl. Mixed finite element approximation of elliptic equations involving high-order derivatives, 2018. download