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In this paper, we propose a new spectral decomposition method to simulate waves propagating in complicated waveguides. For the numerical solutions of waveguide scattering problems, an important task is to approximate the Dirichlet-to-Neumann map efficiently. From previous results, the physical solution can be decomposed into a family of generalized eigenfunctions, thus we can write the Dirichlet-to-Neumann map explicitly by these functions. From the exponential decay of the generalized eigenfunctions, we approximate the Dirichlet-to-Neumann (DtN) map by a finite truncation and the approximation is proved to converge exponentially. With the help of the truncated DtN map, the unbounded domain is truncated into a bounded one, and a variational formulation for the problem is set up in this bounded domain. The truncated problem is then solved by a finite element method. The error estimation is also provided for the numerical algorithm and numerical examples are shown to illustrate the efficiency of the algorithm.

In this paper, we propose a new nonuniform mesh method to simulate acoustic scattering problems in two dimensional periodic structures with non-periodic incident fields numerically. As existing methods are difficult to extend to higher dimensions, we have designed the new method with such extensions in mind. With the help of the Floquet-Bloch transform, the solution to the original scattering problem is written as an integral of a family of quasi-periodic problems. These are defined in bounded domains for each value of the Floquet parameter which varies in a bounded interval. The key step in our method is the numerical approximation of the integral by a quadrature rule adapted to the regularity of the family of quasi-periodic solutions. We design a nonuniform mesh with a Gaussian quadrature rule applied on each subinterval. We prove that the numerical method converges exponentially with respect to both the number of subintervals and the number of Gaussian quadrature points. Some numerical experiments are provided to illustrate the results.

In the author's previous paper (Zhang et al. 2022), exponential convergence was proved for the perfectly matched layers (PML) approximation of scattering problems with periodic surfaces in 2D. However, due to the overlapping of singularities, an exceptional case, i.e., when the wave number is a half integer, has to be excluded in the proof. However, numerical results for these cases still have fast convergence rate and this motivates us to go deeper into these cases. In this paper, we focus on these cases and prove that the fast convergence result for the discretized form. Numerical examples are also presented to support our theoretical results.

The perfectly matched layer (PML) is a very popular tool in the truncation of wave scattering in unbounded domains. In Chandler-Wilde & Monk et al. 2009, the author proposed a conjecture that for scattering problems with rough surfaces, the PML converges exponentially with respect to the PML parameter in any compact subset. In the author's previous paper (Zhang et al. 2022), this result has been proved for periodic surfaces in two dimensional spaces, when the wave number is not a half integer. In this paper, we prove that the method has a high order convergence rate in the 3D bi-periodic surface scattering problems. We extend the 2D results and prove that the exponential convergence still holds when the wavenumber is smaller than $0.5$. For lareger wavenumbers, although exponential convergence is no longer proved, we are able to prove that a higher order convergence for the PML method.

In this paper, we propose a new numerical method for scattering problems in periodic waveguide, based on the newly established contour integral representation of solutions in a previous paper by the author (see [Zhadf]). For this kind of problems, solutions are obtained via the Limiting Absorption Principle and we all them LAP solutions. Based on the Floquet-Bloch transform and analytic Fredholm theory, an LAP solution could be written explicitly as an integral of quasi-periodic solutions on a contour, which depends on the periodic structure. Compared to previous numerical methods for this kind of problems, we do not need to adopt the LAP for approximation, thus a standard error estimation is easily carried out. Based on this method, we also develop a numerical solver for the halfguide problems. This method is also based on the result from [Zhadf], which shows that any LAP solution of a halfguide problem could be extended into a fullguide problem with a non-vanishing source term(not uniquely!). So first we approximate the source term from the boundary data by a regularization method, and then the LAP solution could be obtained from the corresbonding fullguide problem. At the end of this paper, we also show some numerical results to present the efficiency of our numerical methods.

In this paper, we develop a high order numerical method for the numerical solutions of scattering problems with slightly perturbed periodic surfaces in two dimensional spaces. Based on the regularity property introduced in Part I, the decaying rate of the incident field could be transferred directly to the total field for small perturbations. Thus the finite section method could reach a high accuracy rate. With the help of a modification of the truncated problem, the problem is solved by a finite element method. The convergence of the finite element method is proved and numerical examples have been carried out to show the efficiency of the numerical scheme.

In this paper, we will study the Bloch transformed rough surface scattering problems, and propose a numerical method based on the Bloch transformed problems. Based on the mathematical theory of the scattering problems from locally perturbed periodic surfaces, the same techniques will be applied to the rough surface scattering problems, and an equivalent coupled family of quasi-periodic scattering problems in one periodic cell will be established. The most important result obtained in this paper is on the finite Fourier series approximation of the Bloch transformed field with respect to the quasi-periodicity parameter. It will be proved that the finite series is exactly the Bloch transformed solution corresponds to truncated rough surfaces. Thus the truncation provides a reasonable approximation, and could be applied to the numerical solutions. Based on the approximation, a numerical method is proposed for the rough surface scattering problems. The convergence of the numerical method is proved and illustrated by the numerical experiments. The method provides a completely new perspective for the rough surface scattering problems. There is possibility that some high order method will be developed based on this new method.

Waves scattering from unbounded structures are always complicated problems for numerical simulations. For the case that the non-periodic incident field scattered by (locally perturbed) periodic surfaces, with the help of the Bloch transform, the problem could be solved by some finite element methods, if the incident fields decay at certain rate at the infinity. For faster decaying incident fields, a high order numerical method is also available. However, in these cases, the plain waves, which belong to a very important family of incident fields but do not decay at the infinity, are not included. In this paper, we aim to develop the Bloch transform based standard finite method for this certain case, and then establish the high order method afterwards. Numerical experiments have been carried out for both the standard and high order numerical methods. Based on the algorithms for incident plain waves, we could also extend the numerical methods to more generalized cases when only the not so efficient standard method is available.

Rough surface scattering problems are always very challenging both theoretically and numerically. In this paper, we adopt the Bloch transform and the perturbation theory to investigate a special case, i.e., when the rough surface is a slight perturbation of a periodic one. Based on known results from the Bloch transform, the problem could be written into an equivalent bounded variational problem in higher dimensional spaces. The first step is to consider the non-perturbed problem. From the perturbation theory, the solution could be written as a Neumann series. The second step is to consider the slightly perturbed problems. The key point in the process is the relationship between the Dirichlet-to-Neumann map. With the study between these two operators, it is proved that when the right hand side belongs to a certain space, the problem is equivalent to a modified one. Thus a high regularity result is obtained., Comment: Fundamental mistake in the paper. Equation (65) is wrong, which makes the result in Lemma 22 impossible to be proved. Then the main theorem in Section 5, Theorem 23 and 24 could not be proved

Scattering problems with locally perturbed periodic surfaces have been studied both theoretically and numerically in recent years. In this paper, we will discuss the regularity results of the Bloch transform of the total fields. The idea is inspired by Theorem a in \cite{Kirsc1993}, which considered how the total field depends on the wave numbers and the incident angles, with a family of plain incident fields and a smooth enough periodic surface. We will show that when the incident field satisfies some certain conditions, the Bloch transform of the total field depends analytically on the quasi-periodicities one the straight line $\R$ except for a countable number of points, while near such points, a square-root like singularity exists. We also give some examples to show that the conditions are satisfied by a large number of commonly used incident fields. This result also provides a probability to improve the numerical solution of this kind of problems, which is expected to be discussed in our future papers.