Efficient Solution of Large Sparse Eigenvalue Problems in Microelectronic Simulation

Abstract

We present a new Chebyshev-Arnoldi algorithm for finding the lowest energy eigen-functions of an elliptic operator. The algorithm, which is essentially the same for symmetric, nonsymmetric, and complex nonhermitian matrices, is adapted to a specific problem by two subroutines which encapsulate the problem-specific definition of energy, plus the discretization and matrix-vector multiply routines. We adapt the algorithm to two important problems, the self-consistent Schrodinger-Poisson model of quantum-effect devices, and the vector Helmholtz equation for a dielectric waveguide, addressing other important physical, numerical and computational issues as they arise. An asymptotic convergence estimate is derived which shows the Chebyshev-Arnoldi algorithm to be superior to Chebyshev-preconditioned subspace iteration. We also examine Newton methods for general large sparse eigenvalue problems satisfying the overdamping condition and show how to use sparse iterative solvers more effectively in them.