A posteriori error analysis for the conical diffraction problem

Maxwell's equations for conical diffraction can be reduced to a system of two Helmholtz equations in R2 coupled via quasi-periodic transmission conditions on a set of piecewise smooth interfaces. A finite element formulation of the conical diffraction problem is presented in a bounded domain by introducing the nonlocal boundary operators. An a posteriori error estimate is established when the truncation of the nonlocal boundary operators takes place. As our work in Wang et al. (2015), a duality argument is applied to overcome the difficulty caused by the fact that the truncated pseudo-differential mapping does not converge to the original pseudo-differential mapping in its operator norm. The a posteriori error estimate consists of two parts: finite element discretization error and the truncation error of the nonlocal boundary operators. In particular, the truncation error is exponentially decaying with respect to the truncation parameter.