Adaptive DtN Finite Element Methods For Several Grating Problems

This dissertation addresses adaptive DtN finite element methods for the diffraction grating problem, the chiral grating problem, and the nonlinear grating problem.Firstly, the weak formulations for the several grating problems are presented in a bounded domain by introducing the DtN boundary conditions, respectively. In our approach, no extra artificial layer needs to be imposed to surround the computational domain, which is totally different from the perfectly matched absorbing layers(PML) technique, we only need to truncate the nonlocal DtN boundary operators. Then, finite element formulations with the truncation operators are proposed for solving the several grating problems. It should be noted that the conforming element and the edge element discretizations are respectively used for one-dimensional grating and two-dimensional grating problems, where one-dimensional grating problems include the linear grating, the chiral grating, and the nonlinear grating.Based on the corresponding numerical schemes, the a posteriori error estimates are derived for the several grating problems, respectively. In our method, the truncated DtN mapping does not converge to the original DtN mapping. The a posteriori analysis of the adaptive PML method can not apply directly to our adaptive DtN case, since the fact was used that the DtN mapping of the truncated PML problem converges exponen-tially fast to the original DtN mapping. To overcome this difficulty in our a posteriori error analysis, we develop a duality argument similar to that (or the so called Schatz argument) for the a priori error estimates for indefinite problems, but without assuming more regularity than the weak solution.The a posteriori error estimates of the serval grating problems consist of two part-s: the finite element discretization error and the truncation error of boundary operators which decays exponentially with respect to the truncation parameter. With the help of the a posteriori error estimates, the adaptive DtN finite element strategies are respec-tively established for the one-dimensional linear grating problem and the chiral grating problem, such that the truncation parameter is determined through the truncation er-ror and the mesh elements for local refinements are marked through the finite element discretization error.Finally, numerical examples are included to illustrate the effectiveness and fea-sibility of our proposed adaptive DtN algorithm. Especially, for solving the one-dimensional linear grating problems, our method is comparable to the adaptive PML method in performance of approximation errors.