@article{11170,
  author = {Trygve Nilssen and Gunnar Staff and Kent-Andre Mardal},
  title = {Order Optimal Preconditioners for Fully Implicit Runge-Kutta Schemes Applied to the Bidomain Equations},
  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.},
  year = {2011},
  journal = {Numerical Methods for Partial Differential Equations},
  volume = {27},
  pages = {1290-1312},
  publisher = {Wiley Subscription Services, Inc., A Wiley Company},
  doi = {10.1002/num.20582},
  note = {Listed in annual report 2010. Published online june 2010.},
}