In this paper, we present a multi-level mixed element scheme for the Helmholtz transmission eigenvalue problem on polygonal domains that are not necessarily able to be covered by rectangle grids. We first construct an equivalent linear mixed formulation of the transmission eigenvalue problem and then discretize it with Lagrangian finite elements of low regularities. The proposed scheme admits a natural nested discretization, based on which we construct a multi-level scheme. Optimal convergence rate and optimal com- putational cost can be obtained with the scheme.
A Novel Computational and Analytical Framework for 2D Quasiperiodic Helmholtz Eigenvalue Problems via the Projection Method
School of Mathematics and Shing-Tung Yau Center, Southeast University（东南大学数学系及苏步青中心）；Shanghai Institute for Mathematics and Interdisciplinary Sciences (SIMIS)（上海数学与交叉学科研究院）；Department of Applied Mathematics, National Yang Ming Chiao Tung University（国立阳明交通大学应用数学系）；School of Mathematical Sciences, Nanjing Normal University（南京师范大学数学科学学院）
In this paper, based on a domain decomposition (DD) method, we shall propose an efficient two-level preconditioned Helmholtz-Jacobi-Davidson (PHJD) method for solving the algebraic eigenvalue problem resulting from the edge element approximation of the Maxwell eigenvalue problem. In order to eliminate the components in orthogonal complement space of the eigenvalue, we shall solve a parallel preconditioned system and a Helmholtz projection system together in fine space. After one coarse space correction in each iteration and minimizing the Rayleigh quotient in a small dimensional Davidson space…
Well-posed boundary integral equation formulations and Nyström discretizations for the solution of Helmholtz transmission problems in two-dimensional Lipschitz domains
University of New Hampshire（新罕布什尔大学）；New Jersey Institute of Technology（新泽西理工学院）
In this paper, we consider a cubic $H^2$ nonconforming finite element scheme $B_{h0}^3$ which does not correspond to a locally defined finite element with Ciarlet$'$s triple but admit a set of local basis functions. For the first time, we deduce and write out the expression of basis functions explicitly. Distinguished from the most nonconforming finite element methods, $(δΔ_h\cdot,Δ_h\cdot)$ with non-constant coefficient $δ>0$ is coercive on the nonconforming $B_{h0}^3$ space which makes it robust for numerical discretization. For fourth order eigenvalue problem, the $B_{h0}^3$ scheme can prov…
In this paper we revisit the Restricted Additive Schwarz method for solving discretized Helmholtz problems, using impedance boundary conditions on subdomains (sometimes called ORAS). We present this method in its variational form and show that it can be seen as a finite element discretization of a parallel overlapping domain decomposition method defined at the PDE level. In a fourthcoming paper, the authors have proved certain contractive properties of the error propagation operator for this method at the PDE level, under certain geometrical assumptions. We illustrate computationally that thes…
In this paper we are concerned with restricted additive Schwarz with local impedance transformation conditions for a family of Helmholtz problems in two dimensions. These problems are discretized by the finite element method with conforming nodal finite elements. We design and analyze a new adaptive coarse space for this kind of restricted additive Schwarz method. This coarse space is spanned by some eigenvalue functions of local generalized eigenvalue problems, which are defined by weighted positive semi-definite bilinear forms on subspaces consisting of local discrete Helmholtz-harmonic func…
For the Helmholtz equation posed in the exterior of a Dirichlet obstacle, we prove that if there exists a family of quasimodes (as is the case when the exterior of the obstacle has stable trapped rays), then there exist near-zero eigenvalues of the standard variational formulation of the exterior Dirichlet problem (recall that this formulation involves truncating the exterior domain and applying the exterior Dirichlet-to-Neumann map on the truncation boundary). Our motivation for proving this result is that a) the finite-element method for computing approximations to solutions of the Helmholtz…
We consider the Helmholtz transmission problem with piecewise-constant material coefficients, and the standard associated direct boundary integral equations. For certain coefficients and geometries, the norms of the inverses of the boundary integral operators grow rapidly through an increasing sequence of frequencies, even though this is not the case for the solution operator of the transmission problem; we call this phenomenon that of spurious quasi-resonances. We give a rigorous explanation of why and when spurious quasi-resonances occur, and propose modified boundary integral equations that…
In this paper we discuss different transmission operators for the non-overlapping Schwarz method which are suited for solving the time-harmonic Helmholtz equation in cavities (i.e. closed domains which do not feature an outgoing wave condition). Such problems are heavily impacted by back-propagating waves which are often neglected when devising optimized transmission operators for the Schwarz method. This work explores new operators taking into account those back-propagating waves and compares them with well-established operators neglecting these contributions. Notably, this paper focuses on t…
We discuss a quartic eigenvalue problem arising in the context of an optical waveguiding problem involving atomically thick 2D materials. The waveguide configuration we consider consists of a gradient-index (spatially dependent) dielectric equipped with conducting interior interfaces. This leads to a quartic eigenvalue problem with mixed transverse electric and transverse magnetic modes, and strongly coupled electric and magnetic fields. We derive a weak formulation of the quartic eigenvalue problem and introduce a numerical solver based on a quadratification approach in which the quartic eige…
The Restricted Additive Schwarz method with impedance transmission conditions, also known as the Optimised Restricted Additive Schwarz (ORAS) method, is a simple overlapping one-level parallel domain decomposition method, which has been successfully used as an iterative solver and as a preconditioner for discretized Helmholtz boundary-value problems. In this paper, we give, for the first time, a convergence analysis for ORAS as an iterative solver -- and also as a preconditioner -- for nodal finite element Helmholtz systems of any polynomial order. The analysis starts by showing (for general d…
In this paper we discuss the convergence of state-of-the-art optimized Schwarz transmission conditions for Helmholtz problems defined on closed domains (i.e. setups which do not exhibit an outgoing wave condition), as commonly encountered when modeling cavities. In particular, the impact of back-propagating waves on the Dirichlet-to-Neumann map is analyzed. Afterwards, the performance of the well-established optimized 0th-order, evanescent modes damping, optimized 2nd-order and Padé-localized square-root transmission conditions is discussed.
Convergence of overlapping domain decomposition methods with PML transmission conditions applied to nontrapping Helmholtz problems
University College London（伦敦大学学院）；The Chinese University of Hong Kong, Shenzhen（香港中文大学（深圳））；Shenzhen Research Institute of Big Data（深圳大数据研究院）；University of Bath（巴斯大学）；CNRS and Institut de Mathématiques de Toulouse, UMR5219（法国国家科学研究中心与图卢兹数学研究所；图卢兹大学）；Université de Toulouse
We present a new approach to compute selected eigenvalues and eigenvectors of the two-parameter eigenvalue problem. Our method requires computing generalized eigenvalue problems of the same size as the matrices of the initial two-parameter eigenvalue problem. The method is applicable for right definite problems, possibly after performing an affine transformation. This includes a class of Helmholtz equations when separation of variables is applied. We provide a convergence proof for extremal eigenvalues and empirical evidence along with a local convergence proof for other eigenvalues.
Numerical resolution of exterior Helmholtz problems requires some approach to domain truncation. As an alternative to approximate nonreflecting boundary conditions and invocation of the Dirichlet-to-Neumann map, we introduce a new, nonlocal boundary condition. This condition is exact and requires the evaluation of layer potentials involving the free space Green's function. However, it seems to work in general unstructured geometry, and Galerkin finite element discretization leads to convergence under the usual mesh constraints imposed by Gårding-type inequalities. The nonlocal boundary conditi…
The transmission eigenvalue problem arising from the inverse scattering theory is of great importance in the theory of qualitative methods and in the practical applications. In this paper, we study the transmission eigenvalue problem for anisotropic inhomogeneous media in $Ω\subset \mathbb{R}^d$,(d=2,3). Using the T-coercivity and the spectral approximation theory, we derive an a posteriori estimator of residual type and prove its effectiveness and reliability for eigenfunctions. In addition, we also prove the reliability of the estimator for transmission eigenvalues. The numerical experiments…
The paper presents a finite element scheme for the elastic transmission eigenvalue problem written as a fourth order eigenvalue problem. The scheme uses piecewise cubic polynomials and obtains optimal convergence rate. Compared with other low-degree and nonconforming finite element schemes, the scheme inherits the continuous bilinear form which does not need extra stabilizations and is thus simple to implement.
We present a fast spectral Galerkin scheme for the discretization of boundary integral equations arising from two-dimensional Helmholtz transmission problems in multi-layered periodic structures or gratings. Employing suitably parametrized Fourier basis and excluding Rayleigh-Wood anomalies, we rigorously establish the well-posedness of both continuous and discrete problems, and prove super-algebraic error convergence rates for the proposed scheme. Through several numerical examples, we confirm our findings and show performances competitive to those attained via Nyström methods.
A stabilized version of the fundamental solution method to catch ill-conditioning effects is investigated with focus on the computation of complex-valued elastic interior transmission eigenvalues in two dimensions for homogeneous and isotropic media without voids. Its algorithm can be implemented very shortly and adopts to many similar PDE-based eigenproblems as long as the underlying fundamental solution function can be easily generated. We develop a corroborative approximation analysis which also implicates new basic results for transmission eigenfunctions and present some numerical examples…
The transmission eigenvalue problem arises from the inverse scattering theory for inhomogeneous media and has important applications in many qualitative methods. The problem is posted as a system of two second order partial differential equations and is essentially nonlinear, non-selfadjoint, and of higher order. It is nontrivial to develop effective numerical methods and the proof of convergence is challenging. In this paper, we formulate the transmission eigenvalue problem for anisotropic media as an eigenvalue problem of a holomorphic Fredholm operator function of index zero. The Lagrange f…
$\bf H(\mathrm{curl}^2)$ conforming element for Maxwell's transmission eigenvalue problem using fixed-point approach
Beijing Computational Science Research Center（北京计算科学研究中心）；School of Mathematical Sciences, Guizhou Normal University（贵州师范大学数学科学学院）；Department of Mathematics, Wayne State University（韦恩州立大学数学系）
This paper explores a family of generalized sweeping preconditionners for Helmholtz problems with non-overlapping checkerboard partition of the computational domain. The domain decomposition procedure relies on high-order transmission conditions and cross-point treatments, which cannot scale without an efficient preconditioning technique when the number of subdomains increases. With the proposed approach, existing sweeping preconditioners, such as the symmetric Gauss-Seidel and parallel double sweep preconditioners, can be applied to checkerboard partitions with different sweeping directions (…
Computation of Transmission Eigenvalues for Elastic Waves
Academy of Mathematics and System Sciences, Chinese Academy of Sciences（中国科学院数学与系统科学研究院）；Purdue University（普渡大学）；Michigan Technological University（密歇根理工大学）
We consider GMRES applied to discretisations of the high-frequency Helmholtz equation with strong trapping; recall that in this situation the problem is exponentially ill-conditioned through an increasing sequence of frequencies. Under certain assumptions about the distribution of the eigenvalues, we prove upper bounds on how the number of GMRES iterations grows with the frequency. Our main focus is on boundary-integral-equation formulations of the exterior Dirichlet and Neumann obstacle problems in 2- and 3-d; for these problems, we investigate numerically the sharpness (in terms of dependenc…
In this paper, we are concerned with the weighted plane wave least-squares (PWLS) method for three-dimensional Helmholtz equations, and develop the multi-level adaptive BDDC algorithms for solving the resulting discrete system. In order to form the adaptive coarse components, the local generalized eigenvalue problems for each common face and each common edge are carefully designed. The condition number of the two-level adaptive BDDC preconditioned system is proved to be bounded above by a user-defined tolerance and a constant which is dependent on the maximum number of faces and edges per subd…
Well-conditioned boundary integral equation formulations and Nyström discretizations for the solution of Helmholtz problems with impedance boundary conditions in two-dimensional Lipschitz domains
