SUMMARY We present several preconditioned conjugate gradient methods for the iterative solution of the linear, symmetric systems of equations arising from the nite element method in displacement form, both for the h-version and the p-version. The preconditioners are based on a decomposition of the solution space into overlapping subspaces and solving separately on each subspace. A judicious cho...

A class of one-step methods which obtain a block of r new values at each step are studied. The asymptotic behavior of both implicit and predictor-corrector procedures is examined.

We study the performance of Bacon-Shor codes, quantum subsystem codes which are well suited for applications to fault-tolerant quantum memory because the error syndrome can be extracted by performing two-qubit measurements. Assuming independent noise, we find the optimal block size in terms of the bit-flip error probability pX and the phase error probability pZ , and determine how the probabili...