Winther workshop in numerical methods

The workshop starts on Thursday March 14th at 10:15 in B1119,  located on the 11th floor in the Niels Henrik Abel building at the University of Oslo. 

There will be a small lunch served on the 12th floor.

We will also have a workshop dinner on Thursday the 14th. (More information will come)

Programme

Thursday March 14th

Titles and Abstracts:

Rick Falk: Research with Ragnar.

Abstract: In this talk, I will first present a brief select history of Ragnar's research, demonstrating the diversity of topics he as worked on. I will then concentrate on a few of the key ideas that have been central to his research. showing some ways in which they are connected. The topics will be drawn from preconditioning of discrete approximationsof PDE systems, mixed finite element methods, the finite element exterior calculus, and finite element methods for linear elasticity.

Kent Andre Mardal:  Preconditioning of multi-physics problems.

Abstract: In this talk we will discuss preconditioning of multi-physics problems such as the Darcy-Stokes problem. Efficient preconditioners for both the Stokes and Darcy problems are readily available and there are therefore two main problems that need to be addressed: the interface and the robustness with respect to the various parameters. Both problems will be discussed in light of the ‘operator preconditioning framework’. We will motivate the study by discussing current application to Alzheimer´s disease. ​

Kaibo Hu:  Differential complexes and Poincaré-Korn type inequalities.

Abstract: We construct differential complexes with Sobolev, Besov and Triebel-Lizorkin spaces and derive their cohomology. As a result, we show various Poincaré-Korn type inequalities. The results are based on a generalized Bernstein-Gelfand-Gelfand construction and the estimates for the de Rham complexes by Costabel and McIntosh [Mathematische Zeitschrift, 2010].

Marie Rognes: Nearly incompressible generalized poroelasticity - discretization and preconditioning.

Abstract: In this talk, I'll present and analyze mixed finite element formulations of a general family of quasi-static multiple-network poroelasticity (MPET) equations . The MPET equations describe flow and deformation in an elastic porous medium that is permeated by multiple fluid networks of differing characteristics. The MPET equations represent a generalization of Biot's equations, but have received very little attention from the numerical community (up until now!) I will focus on the nearly incompressible case for which standard mixed finite element discretizations of the MPET equations perform poorly, and propose a new mixed finite element formulation based on introducing an additional total pressure variable [Lee, Piersanti, Mardal, Rognes (2019)]. Our work is much inspired by the work of Lee, Mardal and Winther (2017).

Helge Holden: Operator splitting and convergent schemes for the KdV and Benjamin–Ono equations.

Abstract: We show how to analyze solutions of the Cauchy problem for the KdV equation u_t+uu_x=u_xxx using operator splitting. The method is then applied to derive convergent fully discrete difference schemes for the KdV as well as the Benjamin–Ono equation.

Snorre Christiansen:  Finite element systems for smooth differential forms.

Abstract: The framework of finite element systems was introduced to give a unified treatment of finite element spaces adapted to the differential operators gradient, curl and divergence. The vectorfields are then required to be continuous at interfaces either in tangential or normal directions. However for some equations in elasticity and fluid mechanics, the fields should have a higher regularity, say full continuity of the vectorfields, yet somehow be adapted to the same differential operators. We extend the framework of finite element systems to cover such cases, and provide some examples of new finite element spaces that fit the framework. They are particularly adapted to the Stokes problem and nearly incompressible elasticity. This is joint work with Kaibo Hu.

Anders Szepessy:  Approximation of conservation laws.

Abstract: Approximation of conservation laws will be presented, with a local and global view.

Aslak Tveito: Cell-based mathematical models of the electrophysiology of cardiac tissue.

Abstract: Mathematical models of cardiac tissue are defined in terms of a system of partial differential equations defined on the entire cardiac muscle. In these models, cardiomyocytes are not explicitly represented; rather the models are based on average quantities. In fact, for the most commonly used models the extracellular space, the intercellular space of the cell and the cell membrane are all assumed to exist everywhere in the muscle. This constitutes a colossal simplification of the actual physiology, but it has provided useful insight based on computational analyses. However, the accuracy of averaged models is limited by the length scale of the averaging, and as computing power increases, it is possible to get a closer look at how the electrochemical wave underpinning each heart beat actually traverses the heart-muscle. It is now possible to use sub-cell (physical cell) resolution in the computational method and it is therefore about time to re-introduce the physical cell into the mathematical models.  In this talk we show examples of problems were averaged models cannot be used and we will give some examples of the application of a model where individual cells are represented in the model.

Tom Lyche:  Ragnar Winther: From student to leader.

Abstract:  His professional arc from student to international leader is traced and a snapshot of joint work is presented.  Some pictures capture selected highlights from his career and extensive service to the community.