@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.}, }