An Improved Jacobi-Davidson Method With Multi-Level Startup Procedure

The Jacobi-Davidson method is very attractive for the large-scale finite element simulation of electromagnetic cavity problems, because it just requires approximate solutions to some auxiliary systems of linear equations, which are easily obtained by iterative solvers. Moreover, rates of convergence are quadratic to cubic. In consequence, the time spent within the pre-asymptotic region at the early stages of the iteration may have strong impact on overall performance, particularly when the number of eigenvalues sought is small. Therefore, we propose a startup procedure that utilizes p hierarchical finite element spaces to reduce the cost of bridging the pre-asymptotic region.