Notices of the International Congress of Chinese Mathematicians
Volume 4 (2016)
Eigendecompositions and fast eigensolvers for Maxwell equations
Pages: 46 – 54
In this article we report some recent advances and their extensions in the computation of the band structures of photonic crystals, engineered periodic structures made of two or more materials. Our approach is to deduce explicit eigendecompositions for the discrete single and double curl operators associated with the eigenvalue problems resulting from the discretization of the Maxwell eigenvalue problem. These decompositions can be efficiently computed due to our specific and judicious design of the bases for the range and null spaces for which Fast Fourier Transform (FFT) can be used to accelerate the computation. Furthermore, as a result we can derive the null space free method to avoid the commonly annoying large null space issue when it comes to applying iterative eigenvalue solvers. The reduction of the problem size is often substantial. We will discuss a few signposts that made our approach possible and promising to tackle more complicated problems related to metamaterials.