# On adaptive finite element methods for the simulation of two-dimensional photonic crystals

## Inst. Mathematik

The first part of this paper is devoted to the modeling of wave propagation inside a perfect two-dimensional photonic crystal and the spectral analysis of the resulting eigenvalue problem. In the second part of the paper, we focus on a special case, where the eigenvalue problem is linear and Hermitian. We introduce a residual-based estimator approximating the algebraic error, i.e., the error in the computed eigenvector induced by iterative methods. The estimator for the algebraic error approximates the error in the same norm as the one used in already existing estimators for the discretization error, therefore enabling error balancing between both types of errors, and thus improving the efficiency of the adaptive finite element procedure.