Compatible Discretizations for Partial Differential Equations

CMA, University of Oslo, Norway
September 26-28, 2005.

Here you will find presentation slides from the conference

This workshop is in many respects a follow-up of the IMA "hot topics" workshop,

Compatible spatial discretizations for partial differential equations,

held at Institute for mathematics and its applications (IMA), University of Minnesota, May 11-15, 2004, see http://www.ima.umn.edu/complex/spring/discretization.html

Abstract:

The numerical solution of partial differential equations (PDEs) is a fundamental task in science and engineering. A necessary requirement for a numerical scheme is that the discrete solution is close to the exact solution, i.e., that the scheme converges in a proper norm with sufficient speed. However, in recent years it has become clear that for a number of applications certain structure-preserving discretization schemes are superior to others.

Compatible spatial discretizations can be defined as those that inherit or mimic fundamental properties of the PDE such as topology, conservation, symmetries, positivity and maximum principles. Some of these properties are found among schemes based on: mixed finite elements , mimetic finite differences, control volume methods, discrete differential forms, conservative difference schemes, discrete Helmholtz decompositions , finite integration techniques, staggered grid and dual grid methods, etc.

In a parallel development, so called "geometric'' techniques for time-discretization have been developed and shown to be essential for proper discretization of certain ordinary differential equations (ODEs). For applications in areas like computational chemistry, celestial mechanics and particle accelerators the design of special methods inheriting the variational principles or time-reversing symmetries of the exact solutions are crucial. Proper time-discretization leads to methods that preserve Lie group properties of the solution propagator such as symplecticity, time-reversibility, discrete symmetries, phase space volume, etc.

A fully compatible approach to time-dependent PDEs should take both the space and time discretization into consideration. The aim of this workshop is to serve as a forum where different approaches to compatible discretizations are presented and compared. The attention will be given to the identification of the fundamental structures essential to preserve in a discretization, and the implications of such properties on stability and accuracy.

Invited speakers:

• Ivar Aavatsmark, University of Bergen
• Douglas N. Arnold, University of Minnesota
• Pavel Bochev, Sandia National Laboratories
• Franco Brezzi, University of Pavia
• Richard S. Falk, Rutgers University
• Claes Johnson, Chalmers, Gøteborg University
• Hans Munthe Kaas, University of Bergen
• Christian Lubich, University of Tubingen
• Donatella Marini, University of Pavia
• Jean-Claude Nédélec, Ecole Polytechnique
• Brynjulf Owren, Norwegian University of Science and Technology
• Sebastian Reich, Potsdam University
• Joachim Schöberl, RICAM Linz
• Ivan Yotov, University of Pittsburgh

Participants

A full list of participants can be found here

Program (subject to change):

Invited presentations are 50 minutes with an additional 10 minutes of discussion. Contributed Presentations are 25 minutes with an additional 5 minutes of discussion.

Monday, Sep 26:

10.00-10.05 Ragnar Winther, Director CMA, Opening
10.05-11.05 Douglas N. Arnold, University of Minnesota:
Differential complexes and mixed finite elements for elasticity. Abstract.
11.05-11.30 Coffee
11.30-12.30 Christian Lubich, University of Tubingen:
Variational approximations in quantum molecular dynamics. Abstract.

12.30-14.00 Lunch

14.00-15.00 Ivar Aavatsmark, University of Bergen
Control-volume discretizations compatible with physics. Abstract.
14.00-15.30 Elizabeth Mansfield, University of Kent:
Noether's Theorem for smooth, finite difference and finite element methods. Abstract.
15.30-16.00 Coffee
16.00-17.00 Sebastian Reich, Potsdam University: Multi-scale and conservative time-stepping methods for numerical weather prediction. Abstract.

Tuesday, Sep 27

09.00-10.00 Franco Brezzi, University of Pavia: New development in mimetic finite difference methods.
10.00-10.30 Coffee
10.30-11.30 Pavel Bochev, Sandia National Laboratories:
Mimetic Discrete Models with Weak Material Laws, or Least Squares Principles Revisted. Abstract.
11.30-12.30 Hans Munthe Kaas, University of Bergen: On Group Fourier Analysis and Symmetry Preserving Discretizations of PDEs. Abstract.

12.30-14.00 Lunch

14.00-15.00 Richard S. Falk, Rutgers University:
Mixed Finite Elements for Elasticity - a Constructive Approach. Abstract.
15.00-15.30 Donatella Marini, University of Pavia: Stabilization mechanisms in discontinuous Galerkin methods.
15.30-16.00 Coffee
16.30-17.30 Joachim Schoberl, RICAM Linz:
Commuting quasi-interpolation operators for H(curl) and H(div). Abstract.

1900 -> Dinner (information and registration during Monday)

Wednesday, Sep 28

09.00-10.00 Jean-Claude Nédélec, Ecole Polytechnique: Preconditioning Harmonic Maxwell Integral Equation. Abstract.
10.00-10.30 Coffee
10.30-11.00 Snorre Chrstiansen, CMA:
On a dual finite element complex. Abstract.
11.00-12.00 Brynjulf Owren, Norwegian University of Science and Technology: Lie Group Integrators. Abstract.

12.00-13.30 Lunch

13.30-14.30 Ivan Yotov, University of Pittsburgh:
Relationships between mixed finite element methods, multipoint flux approximation methods, and mimetic finite difference methods. Abstract.
14.30-15.00 Coffee
15.00-16.00 Claes Johnson, Chalmers, Gøteborg University
Irreversibility in Reversible Systems. Abstract.

Site for workshop, travel description:

CMA / University of Oslo. Visiting address: Moltke Moes vei 35, (Mathematics building), no 14 on this map. More detailed travelling description here.

The conference will take place in 12th floor.

Those who will stay at Hotel Gyldenløve, tram line 11 and 19 are passing the hotel in Bogstadveien 20. See tram map here (leave at 'Vibes gate' stop)

Registration:

No fee will be charged for participating in the conference. We are deliberately trying to keep this workshop at a moderate size, with approximately 40 participants. To request registration send an email to:
Runhild Aae Klausen (runhildk 'at' ifi.uio.no)
by August 25.

Organizing committee:

• Snorre Christiansen, CMA,
• Runhild Aae Klausen, CIPR/CMA, and
• Per Christian Moan, CMA
• Ragnar Winther, CMA

Sponsors: