Numerical Methods for Partial Differential Equations 2
University of Vienna, Wintersemester 2014

The course will focus on finite element approximation of problems in mixed variational formulation.

Aim: Presenting theoretical and numerical aspects of Mixed Finite Element Methods for the numerical approximation of Partial Differential Equations arising from different applications.

Schedule: Tue 12:15-13:45 SR12, Oskar-Morgenstern-Platz 1
First Lecture: Tue October 7, 2014

- Example of problems in mixed variational formulation
- FEM discretization of mixed problems: general theory
- The Stokes problem
- The Darcy problem
- If time permits, other applications (Maxwell, linear elasticity).

Suggested reading:
D. Boffi, F. Brezzi, M. Fortin, Mixed Finite Element Methods and Applications, Springer, 2013.
A. Quarteroni, A. Valli, Numerical Approximation of Partial Differential Equations, Springer, 2008.
Other material will be suggested during the course.

Evaluation: Final exam

07/10/2014: Example of problems in mixed formulation: the Darcy and the Stokes problem
14/10/2014: Abstract mixed variational formulation; interpretation as a Lagrange multiplier method for a constrained minimization problem; Banach's closed range theorem
21/10/2014: Well-posedness of mixed variational formulations (conditions and proofs)
28/10/2014: Inf-sup condition; well-posedness of Dirichlet-Darcy and of Stokes
04/11/2014: Analysis of discrete mixed formulations (ellipticity on the discrete kernel and discrete inf-sup condition)
11/11/2014: Analysis of discrete mixed formulations: conclusion; Fortin's trick; Clément's operator
18/11/2014: Discretisation of the Stokes problem; locking phenomenon, spurious pressure modes; P1-P0 is not inf-sup stable
25/11/2014: P2-P0 is inf-sup stable; reduced P2-P0, Q2-P0, reduced Q2-P0, serendipity element
02/12/2014: Q1-P0 is not inf-sup stable; making use of internal d.o.f.: case of continuous pressure elements (MINI)
09/12/2014: Case of discontinuous pressure elements (Crouzeix-Raviart, Q2-P1); numerical testing of the inf-sup condition
16/12/2014: Numerical testing of the inf-sup condition (conclusion); finite elements for the Darcy problem