Accuracy and performance issues of spectral preconditioners in semiconductor device simulation

In most numerical simulations in computational science and engineering, a preconditioner is mandatory to iteratively solve large sparse linear systems. In theory, a good preconditioner clusters all eigenvalues of the iteration matrix around one. However, in practice, there may still be a few outliers. In particular, small eigenvalues deteriorate the convergence speed and may affect the ultimate accuracy. Spectral preconditioners can be used to circumvent these problems. They are constructed with the help of an invariant subspace that contains the eigenvectors corresponding to the small eigenvalues. In this paper, we investigate spectral preconditioners in the field of semiconductor device simulation. We look into different formulations and their influence on the accuracy. Performance issues of spectral preconditioners are the second topic we investigate.