Numerical Methods for Partial Differential
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).
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