@article {Simula.SC.35,
	title = {Order Optimal Preconditioners for Fully Implicit Runge-Kutta Schemes Applied to the Bidomain Equations},
	journal = {Numerical Methods for Partial Differential Equations},
	volume = {27},
	year = {2011},
	note = {Listed in annual report 2010. Published online june 2010.},
	pages = {1290-1312},
	publisher = {Wiley Subscription Services, Inc., A Wiley Company},
	abstract = {The partial differential equation part of the bidomain equations is discretized in time with fully implicit Runge-Kutta methods, and the resulting block systems are preconditioned with a block diagonal preconditioner. By studying the time-stepping operator in the proper Sobolev spaces, we show that the preconditioned systems have bounded condition numbers given that the Runge-Kutta scheme is A-stable and irreducible with an invertible coefficient matrix. A new proof of order optimality of the preconditioners for the one-leg discretization in time of the bidomain equations is also presented. The theoretical results are verified by numerical experiments. Additionally, the concept of weakly positive-definite matrices is introduced and analyzed.},
	doi = {10.1002/num.20582},
	author = {Nilssen, Trygve Kastberg and Gunnar Andreas {Staff} and Mardal, Kent-Andre}
}
@article {Mardal.2006.5,
	title = {Order Optimal Preconditioners for Implicit Runge-Kutta Schemes Applied to Parabolic PDE{\textquoteright}s},
	journal = {SIAM Journal on Scientific Computing},
	volume = {29},
	number = {1},
	year = {2007},
	note = {Accepted for publication at SIAM Journal of Scientific Computing},
	pages = {361-375},
	abstract = {In this paper we show that standard preconditioners for parabolic PDEs discretized by implicit Euler or Crank-Nicolson schemes can be reused for higher-order fully implicit Runge-Kutta time discretization schemes. We prove that the suggested block diagonal preconditioners are order-optimal for A-stable and irreducible Runge-Kutta schemes with invertible coefficient matrices. The theoretical investigations are confirmed by numerical experiments.},
	author = {Mardal, Kent-Andre and Nilssen, Trygve Kastberg and Gunnar Andreas {Staff}}
}
@article {Mardal.2006.7,
	title = {Using Python to Solve Partial Differential Equations},
	journal = {Computing in Science \& Engineering},
	volume = {9},
	number = {3},
	year = {2007},
	month = {May/June},
	pages = {48-51},
	author = {Mardal, Kent-Andre and Skavhaug, Ola and Glenn T. {Lines} and Gunnar Andreas {Staff} and {\O}deg{\r a}rd, {\r A}smund}
}
@inproceedings {Staff.2006.2,
	title = {Block Preconditioners for Fully Implicit Runge-Kutta Schemes Applied to the Bidomain Equations},
	journal = {ECCOMAS CFD 06},
	year = {2006},
	note = {The proceeding is published electronically},
	pages = {1-15},
	publisher = {European Community on Computational Methods in Applied Sciences},
	type = {Conference},
	abstract = {Recently, the authors presented different block preconditioners for implicit Runge-Kutta discretization of the heat equation. The preconditioners were block Jacobi and block Gauss-Seidel preconditoners where the blocks reused existing preconditioners for the implicit Euler discretization of the same equation. In this paper we will introduce similar block preconditioners for the implicit Runge-Kutta discretization of the Bidomain equation. We will, by numerical experiments, show the properties of the preconditoners, and that higher-order Runge-Kutta discretization of the Bidomain equation may be superior to lower-order in some cases.},
	isbn = {90-9020970-0},
	author = {Gunnar Andreas {Staff} and Mardal, Kent-Andre and Nilssen, Trygve Kastberg},
	editor = {Wesseling, P. and O{\~n}ate, E. and P{\'e}riaux, J.}
}
@phdthesis {Staff.2006.3,
	title = {Numerical Computation of Initial Value Problems Originating From Partial Differential Equations},
	year = {2006},
	school = {University of Oslo},
	type = {phd},
	author = {Gunnar Andreas {Staff}}
}
@article {Maday.2006.1,
	title = {The Parareal-in-Time Algorithm: Basics, Stability Analysis and More},
	journal = {journal},
	year = {2006},
	abstract = {The parareal-in-time algorithm allows to take benefit of a parallel architecture of large scale computing resources in order to decrease the restitution time for the numerical simulation of time dependent problems. The method can be presented as a predictor corrector scheme where the predictor is based on a coarse grain and inexpensive simulation, solved in a serial manner and the expensive corrector can be spread on different processors. Like for domain decomposition techniques, the algorithm is based on the decomposition of the global simulation time interval into slabs, each slab being dedicated to a processor. After reminding the basics of the approximation, we discuss the stability of the algorithm for a system of autonomous differential equations. The stability function for the algorithm is derived and analyzed, based on various choices of schemes in time for the coarse and the fine propagator. We then present some complementary analysis that provides the frame for the definition of the cheap coarse simulation. Finally, numerical results for the viscous Burger{\textquoteright}s equation are presented.},
	author = {Maday, Yvon and Einar M. {R{\o}nquist} and Gunnar Andreas {Staff}}
}
@article {Staff.2006.1,
	title = {Preconditioning of Fully Implicit Runge-Kutta Schemes for Parabolic PDEs},
	journal = {Model. Identif. Control},
	volume = {27},
	number = {2},
	year = {2006},
	note = {An earlier version of this paper var publisehd in SIMS05. 46th Conference on Simulation and Modeling, Trondheim, October 13-14, edited by J{\o}rn Amundsen, Helge I. Andersson, Elena Celledoni, Tommy Gravdahl, Finn A. Michelsen, Henrik R. Nagel, Thorvald Natvig. Tapir Academic Press, pages 315-324, 2005.},
	pages = {109-123},
	abstract = {Recently, the authors introduced a preconditioner for the linear systems that arise from fully implicit Runge-Kutta time stepping schemes applied to parabolic PDEs \cite{mns}. The preconditioner was a block Jacobi preconditioner, where each of the blocks were based on standard preconditioners for low-order time discretizations like implicit Euler or Crank-Nicolson. It was proven that the preconditioner is optimal with respect to the timestep and the discretization parameter in space. In this paper we will improve the convergence by considering other preconditioners like the upper and the lower block Gauss-Seidel preconditioners, both in a left and right preconditioning setting. Finally, we improve the condition number by using a generalized Gauss-Seidel preconditioner.},
	author = {Gunnar Andreas {Staff} and Mardal, Kent-Andre and Nilssen, Trygve Kastberg}
}
@inproceedings {Staff.2005.1,
	title = {The Accuracy of the Velocity Approximation With Respect to the Pressure Approximation for Incompressible Fluid Flow},
	journal = {Third National Conference on Computational Mechanics (MekIT 05)},
	year = {2005},
	month = {May},
	pages = {297-311},
	publisher = {Tapir Academic Press},
	type = {1},
	edition = {first},
	address = {Trondheim, May 11-12},
	abstract = {We consider the accuracy of the velocity approximation with respect to the pressure approximation using various standard finite element methods for incompressible fluid flow. Of particular interest is the case where the hydrostatic part of the pressure is dominating, which implies that the pressure may be large relative to the velocity. Some standard mixed elements like the Mini element and the Taylor-Hood element handle this case well, while others like the P\_2-P\_0 element and the Crouzeix-Raviart element do not. This seems to be related to the fact that the Taylor-Hood and the Mini elements yield a stable approximation of the mixed Poisson problem.},
	author = {Gunnar Andreas {Staff} and Mardal, Kent-Andre},
	editor = {Bj{\o}rn H. {Skallerud} and Helge I. {Andersson}}
}
@inproceedings {Staff.2005.2,
	title = {Preconditioning of Fully Implicit Runge-Kutta Schemes for Parabolic PDEs},
	journal = {SIMS05. 46th Conference on Simulation and Modeling, Trondheim, October 13-14},
	year = {2005},
	pages = {315-324},
	publisher = {Tapir Academic Press},
	type = {1},
	abstract = {Recently, the authors introduced a preconditioner for the linear systems that arise from fully implicit Runge-Kutta time stepping schemes applied to parabolic PDEs \cite{mns}. The preconditioner was a block Jacobi preconditioner, where each of the blocks were based on standard preconditioners for low-order time discretizations like implicit Euler or Crank-Nicolson. It was proven that the preconditioner is optimal with respect to the timestep and the discretization parameter in space. In this paper we will improve the convergence by considering other preconditio ners like the upper and the lower block Gauss-Seidel preconditioners, both in a left and right preconditioning setting . Finally, we improve the condition number by using a generalized Gauss-Seidel p reconditioner.},
	author = {Gunnar Andreas {Staff} and Mardal, Kent-Andre and Nilssen, Trygve Kastberg},
	editor = {Amundsen, J{\o}rn and Helge I. {Andersson} and Celledoni, Elena and Gravdahl, Tommy and Finn A. {Michelsen} and Henrik R. {Nagel} and Natvig, Thorvald}
}
@inbook {SC.5.Staff.2003,
	title = {Stability of the Parareal Algorithm},
	booktitle = {Domain Decomposition Methods in Science and Engineering},
	series = {Lecture Notes in Computational Science and Engineering},
	volume = {40},
	year = {2003},
	pages = {449-454},
	publisher = {Springer},
	organization = {Springer},
	author = {Gunnar Andreas {Staff} and Einar M. {R{\o}nquist}},
	editor = {Kornhuber, R. and Hoppe, R.H.W. and Keyes, D.E. and P{\'e}riaux, J. and Pironneau, O. and Xu, J.}
}
