A two-level preconditioned Helmholtz-Jacobi-Davidson method for the Maxwell eigenvalue problem

In this paper, based on a domain decomposition method, we propose an efficient two-level preconditioned Helmholtz-Jacobi-Davidson (PHJD) method for solving the algebraic eigenvalue problem resulting from edge element approximation of Maxwell problem. order to eliminate components in orthogonal complement space eigenvalue, shall solve parallel system and Helmholtz projection together fine space. After one coarse correction each iteration minimizing Rayleigh quotient small dimensional Davidson space, finally get error reduction PHJD as γ = c ( H stretchy="false">) 1 − C δ<!-- δ <mml:mrow class="MJX-TeXAtom-ORD"> 2 \gamma =c(H)(1-C\frac {\delta ^{2}}{H^{2}}) , where alttext="upper C"> encoding="application/x-tex">C is constant independent mesh size alttext="h"> h encoding="application/x-tex">h diameter subdomains H"> encoding="application/x-tex">H alttext="delta"> encoding="application/x-tex">\delta overlapping among subdomains, alttext="c encoding="application/x-tex">c(H) decreasing monotonically alttext="1"> encoding="application/x-tex">1 right-arrow 0"> stretchy="false">→<!-- → <mml:mn>0 encoding="application/x-tex">H\to 0 which means greater number better convergence rate. Numerical results supporting our theory are given.