An adaptive finite volume method for the diffraction grating problem with the truncated DtN boundary condition

In this paper, we develop a adaptive finite volume method with the truncation of the nonlocal boundary operators for the wave scattering by periodic structures. The related truncation parameters are chosen through sharp a posteriori error estimate of the finite volume method. The crucial part of the a posteriori error analysis is to develop a duality argument technique and use a L2-orthogonality property of the residual which plays a similar role as the Galerkin orthogonality. The a posteriori error estimate consists of two parts, the finite volume discretization error for adapting meshes and the truncation error of boundary operators which decays exponentially with respect to the truncation parameter N. Numerical experiments are presented to confirm our theoretical analysis and show the efficiency and robustness of the proposed adaptive method.