Computing extremal eigenvalues for three-dimensional photonic crystals with wave vectors near the Brillouin zone center
The authors consider a matrix eigenvalue problem \(Ax=\lambda Bx\) with \(A\) positive semi-definite and \(B\) positive definite and diagonal, arising in the finite difference solution of Maxwell's equations. The problem studied arises in the study of wave vectors close to the centre of the Brillouin zone of a cubic photonic crystal (a periodic dielectric medium). This problem has many zero eigenvalues, a couple of very small non-zero eigenvalues, and several much larger eigenvalues. The authors determine the number of zero eigenvalues and analyze the structure of the null spaces. They compare two methods (the hybrid Jacobi-Davidson method and the shift-and-invert Krylov-Schur method) for computing the small non-zero eigenvalues, and numerically test how performance is affected by the choice of shift values, condition numbers, and initial vectors.
scientific article; zbMATH DE number 6176968
scientific article