Parallel two-level domain decomposition based Jacobi–Davidson algorithms for pyramidal quantum dot simulation

We consider a quintic polynomial eigenvalue problem arising from the finite volume discretization of a quantum dot simulation problem. The problem is solved by the Jacobi–Davidson (JD) algorithm. Our focus is on how to achieve the quadratic convergence of JD in a way that is not only efficient but also scalable when the number of processor cores is large. For this purpose, we develop a projected two-level Schwarz preconditioned JD algorithm that exploits multilevel domain decomposition techniques. The pyramidal quantum dot calculation is carefully studied to illustrate the efficiency of the proposed method. Numerical experiments confirm that the proposed method has a good scalability for problems with hundreds of millions of unknowns on a parallel computer with more than 10,000 processor cores.