**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

**Outline:**

- 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

**Program:**

**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

**13/01/2014:**

**20/01/2014:**

**27/01/2014:**