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FEM and CIP-FEM for Helmholtz Equation with High Wave Number and PML truncation
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Abstract
The Helmholtz scattering problem with high wave number is truncated by the perfectly matched layer (PML) technique and then discretized by the linear continuous interior penalty finite element method (CIP-FEM). It is proved that the truncated PML problem satisfies the inf--sup condition with inf--sup constant of order . Stability and convergence of the truncated PML problem are discussed. In particular, the convergence rate is twice of the previous result. The preasymptotic error estimates in the energy norm of the linear CIP-FEM as well as FEM are proved to be under the mesh condition that is sufficiently small. Numerical tests are provided to illustrate the preasymptotic error estimates and show that the penalty parameter in the CIP-FEM may be tuned to reduce greatly the pollution error.
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We study the coercivity properties and the norm dependence on the wavenumber k of certain regularized combined field boundary integral operators that we recently introduced for the solu-tion of two and three-dimensional acoustic scattering problems with Neumann boundary condi-tions. We show that in the case of circular and spherical boundaries, our regularized combined field boundary integral operators are coercive for large enough values of the coupling parame-ter, and that the norms of these operators are bounded by constant multiples of the coupling parameter. We establish that the norms of the regularized combined field boundary integral op-erators grow modestly with the wavenumber k for smooth boundaries and we provide numerical evidence that these operators are coercive for two-dimensional starlike boundaries. We present and analyze a fully discrete collocation (Nyström) method for the solution of two-dimensional acoustic scattering problems with Neumann boundary conditions based on regularized combined field integral equations. In particular, for analytic boundaries and boundary data, we establish pointwise superalgebraic convergence rates of the discrete solutions.
A preasymptotic error analysis of the finite element method (FEM) and some
continuous interior penalty finite element method (CIP-FEM) for Helmholtz
equation in two and three dimensions is proposed. - and - error
estimates with explicit dependence on the wave number k are derived. In
particular, it is shown that if is sufficiently small, then
the pollution errors of both methods in -norm are bounded by
, which coincides with the phase error of the FEM obtained
by existent dispersion analyses on Cartesian grids, where h is the mesh size,
p is the order of the approximation space and is fixed. The CIP-FEM extends
the classical one by adding more penalty terms on jumps of higher (up to p-th
order) normal derivatives in order to reduce efficiently the pollution errors
of higher order methods. Numerical tests are provided to verify the theoretical
findings and to illustrate great capability of the CIP-FEM in reducing the
pollution effect.
A numerical method for approximating a pseudodifferential system describing attenuated, scalar waves is introduced and analyzed. Analytic properties of the solutions of the pseudodifferential systems are determined and used to show convergence of the numerical method. Experiments using the method are reported.
We present some new a priori estimates of the solutions to three-dimensional elliptic interface problems and static Maxwell interface system with variable coefficients. Different from the classical a priori estimates, the physical coefficients of the interface problems appear in these new estimates explicitly.
In this article we study the mesh termination method in computational scattering theory known as the method of Perfectly Matched
Layer (PML). This method is based on the idea of surrounding the scatterer and its immediate vicinity with a fictitious absorbing
non-reflecting layer to damp the echoes coming from the mesh termination surface. The method can be formulated equivalently
as a complex stretching of the exterior domain. The article is devoted to the existence and convergence questions of the solutions
of the resulting equations. We show that with a special choice of the fictitious absorbing coefficient, the PML equations
are solvable for all wave numbers, and as the PML layer is made thicker, the PML solution converge exponentially towards the
actual scattering solution. The proofs are based on boundary integral methods and a new type of near-field version of the
radiation condition, called here the double surface radiation condition.
In this paper, which is part II in a series of two, the pre-asymptotic error
analysis of the continuous interior penalty finite element method (CIP-FEM) and
the FEM for the Helmholtz equation in two and three dimensions is continued.
While part I contained results on the linear CIP-FEM and FEM, the present part
deals with approximation spaces of order . By using a modified duality
argument, pre-asymptotic error estimates are derived for both methods under the
condition of , where
k is the wave number, h is the mesh size, and is a constant
independent of , and the penalty parameters. It is shown that the
pollution errors of both methods in -norm are if
p=O(1) and are
if the exact solution u\in H^2(\Om) which coincide with existent dispersion
analyses for the FEM on Cartesian grids. Here \si is a constant independent
of , and the penalty parameters. Moreover, it is proved that the
CIP-FEM is stable for any and penalty parameters with positive
imaginary parts. Besides the advantage of the absolute stability of the CIP-FEM
compared to the FEM, the penalty parameters may be tuned to reduce the
pollution effects.
We consider absorbing layers that are extensions of the PML of Berenger (1994). These will be constructed both for time problems and for Helmholtz-like equations. We shall consider applications to electricity and magnetism and acoustics (with a mean flow) in both physical space and in the time Fourier space (Helmholtz equation). Numerical results are presented showing the efficiency of this condition for the time dependent Maxwell equations.
An adaptive perfectly matched layer (PML) technique for solv- ing the time harmonic electromagnetic scattering problems is developed. The PML parameters such as the thickness of the layer and the fictitious medium property are determined through sharp a posteriori error estimates. Combined with the adaptive finite element method, the adaptive PML technique provides a complete numerical strategy to solve the scattering problem in the frame- work of FEM which produces automatically a coarse mesh size away from the fixed domain and thus makes the total computational costs insensitive to the thickness of the PML absorbing layer. Numerical experiments are included to illustrate the competitive behavior of the proposed adaptive method.
An edge element adaptive strategy with error control is developed for wave scattering by biperiodic structures. The unbounded computational domain is truncated to a bounded one by a perfectly matched layer (PML) technique. The PML parameters, such as the thickness of the layer and the medium properties, are determined through sharp a posteriori error estimates. Numerical experiments are presented to illustrate the competitive behavior of the proposed adaptive method.
In this paper, we propose a uniaxial perfectly matched layer (PML) method for solving the time-harmonic scattering problems in two-layered media. The exterior region of the scatterer is divided into two half spaces by an infinite plane, on two sides of which the wave number takes different values. We surround the computational domain where the scattering field is interested by a PML with the uniaxial medium property. By imposing homogeneous boundary condition on the outer boundary of the PML, we show that the solution of the PML problem converges exponentially to the solution of the original scattering problem in the computational domain as either the PML absorbing coefficient or the thickness of the PML tends to infinity.
This paper develops and analyzes some interior penalty discontinuous Galerkin methods using piecewise linear polynomials for the Helmholtz equation with the first order absorbing boundary condition in the two and three dimensions. It is proved that the proposed discontinuous Galerkin methods are stable (hence well-posed) without any mesh constraint. For each fixed wave number k, optimal order (with respect to h) error estimate in the broken -norm and sub-optimal order estimate in the -norm are derived without any mesh constraint. The latter estimate improves to optimal order when the mesh size h is restricted to the preasymptotic regime (i.e., ). Numerical experiments are also presented to gauge the theoretical result and to numerically examine the pollution effect (with respect to k) in the error bounds. The novelties of the proposed interior penalty discontinuous Galerkin methods include: first, the methods penalize not only the jumps of the function values across the element edges but also the jumps of the normal and tangential derivatives; second, the penalty parameters are taken as complex numbers of positive imaginary parts so essentially and practically no constraint is imposed on the penalty parameters. Since the Helmholtz problem is a non-Hermitian and indefinite linear problem, as expected, the crucial and the most difficult part of the whole analysis is to establish the stability estimates (i.e., a priori estimates) for the numerical solutions. To the end, the cruxes of our analysis are to establish and to make use of a local version of the Rellich identity (for the Laplacian) and to mimic the stability analysis for the PDE solutions given in \cite{cummings00,Cummings_Feng06,hetmaniuk07}.
RESUMEN RESUMEN
This paper is concerned with convergence analysis of the perfectly matched layer (PML) problem in spherical coordinates for the three -dimensional electromagnetic scattering . Under some simple assumptions on the PML medium parameter , it is shown that the truncated PML problem attains a unique solution . The main result of the paper is to establish an explicit error estimate between the solution of the scattering problem and that of the truncated PML problem . The error estimate implies , in particular, that the PML solution converges exponentially to the scattering solution by increasing either the PML medium parameter or the PML layer thickness . The convergence result is expected to be useful for determining the PML medium parameter in the computational electromagnetic scattering problems .
An error analysis is presented for the spectral-Galerkin method to the Helmholtz equation in 2- and 3-dimensional exterior domains. The problem in unbounded domains is first reduced to a problem on a bounded domain via the Dirichlet-to-Neumann operator, and then a spectral-Galerkin method is employed to approximate the reduced problem. The error analysis is based on exploring delicate asymptotic behaviors of the Hankel functions and on deriving a priori estimates with explicit dependence on the wave number for both the continuous and the discrete problems. Explicit error bounds with respect to the wave number are derived, and some illustrative numerical examples are also presented.
In this paper we consider the problem of scattering of time-harmonic acoustic waves by a bounded sound soft obstacle in two and three dimensions, studying dependence on the wave number in two classical formulations of this problem. The first is the standard variational/weak formulation in the part of the exterior domain contained in a large sphere, with an exact Dirichlet- to-Neumann map applied on the boundary. The second formulation is as a second kind boundary integral equation in which the solution is sought as a combined single- and double-layer potential. For the variational formulation we obtain, in the case when the obstacle is starlike, explicit upper and lower bounds which show that the inf-sup constant decreases like k 1 as the wave number k increases. We also give an example where the obstacle is not starlike and the inf-sup constant decreases at least as fast as k 2. For the boundary integral equation formulation, if the boundary is also Lipschitz and piecewise smooth, we show that the norm of the inverse boundary integral operator is bounded independently of k if the coupling parameter is chosen correctly. The methods we use also lead to explicit bounds on the solution of the scattering problem in the energy norm when the obstacle is starlike in which the dependence of the norm of the solution on the wave number and on the geometry are made explicit.
: In 1994 B'erenger showed how to construct a perfectly matched absorbing layer for the Maxwell system in rectilinear coordinates. This layer absorbs waves of any wave-length and any frequency without reflection and thus can be used to artificially terminate the domain of scattering calculations. In this paper we show how to derive and implement the B'erenger layer in curvilinear coordinates (in two space dimensions). We prove that an infinite layer of this type can be used to solve time harmonic scattering problems. We also show that the truncated B'erenger problem has a solution except at a discrete set of exceptional frequencies (which might be empty). Finally numerical results show that the curvilinear layer can produce accurate solutions in the time and frequency domain. Key-words: Perfectly Matched Layer, computational electromagnetics, Absorbing layers (R'esum'e : tsvp) Research funded in part by a grant from AFOSR, USA. This paper has been submited to SIAM Scientific Computin...
In this paper we study the PML method for Helmholtz-type scattering problems with radially symmetric potential. The PML method consists in surrounding the computational domain by a Perfectly Matched sponge Layer. We prove that the approximate solution obtained by the PML method converges exponentially fast to the true solution in the computational domain as the thickness of the sponge layer tends to infinity. This is a generalization of results by Lassas and Somersalo based on boundary integral equation techniques. Here we use techniques based on the pole condition instead. This makes it possible to treat problems without an explicitly known fundamental solution
This article is devoted to wavenumber explicit analysis of the electric field satisfying the second-order time-harmonic Maxwell equations in a spherical shell and, hence, for variant scatterers with ϵ-perturbation of the inner ball radius. The spherical shell model is obtained by assuming that the forcing function is zero outside a circumscribing ball and replacing the radiation condition with a transparent boundary condition involving the capacity operator. Using the divergence-free vector spherical harmonic expansions for two components of the electric field, the Maxwell system is reduced to two sequences of decoupled one-dimensional boundary value problems in the radial direction. The reduced problems naturally allow for truncated vector spherical harmonic spectral approximation of the electric field and one-dimensional global polynomial approximation of the boundary value problems. We analyse the error in the resulting spectral approximation for the spherical shell model. Using a perturbation transformation, we generalize the approach for ϵ-perturbed nonspherical scatterers by representing the resulting field in ϵ-power series expansion with coefficients being spherical shell electric fields.
Existence uniqueness and error estimates for Ritz Galerkin methods are discussed in the case where the associated bilinear form satisfies a Carding type inequality, i.e., it is indefinite in a certain way. An application to the finite element method is given.
The development of numerical methods for solving the Helmholtz equation, which behaves robustly with respect to the wave number, is a topic of vivid research. It was observed that the solution of the Galerkin finite element method (FEM) differs significantly from the best approximation with increasing wave number. Many attempts have been presented in the literature to eliminate this lack of robustness by various modifications of the classical Galerkin FEM. However, we will prove that, in two and more space dimensions, it is impossible to eliminate this so-called pollution effect. In contrast, we will present a generalized FEM in one dimension that behaves robustly (i.e., is pollution-free) with respect to the wave number. The theory developed in this paper can also be used for the comparison of different discretization methods with respect to the size of their pollution.
Ever since the pioneering work of Ihlenburg and Babuška for the one-dimensional Helmholtz equation (1995, Comput. Math. Appl., 30, 9-37), a lot of pre-asymptotic analyses, including dispersion and pollution analyses, have been carried out for finite element methods (FEMs) for the Helmholtz equation, but no rigorous pre-asymptotic error analysis of the FEM for the Helmholtz equation in two and three dimensions has been done yet. This paper addresses pre-asymptotic stability and error estimates of some continuous interior penalty finite element methods (CIP-FEM) and the FEM using piecewise linear polynomials for the Helmholtz equation in two and three dimensions. The CIP-FEM, which was first proposed by Douglas and Dupont for elliptic and parabolic problems in the 1970s and then successfully applied to convection-dominated problems as a stabilization technique, modifies the bilinear form of the FEM by adding a least squares term penalizing the jump of the gradient of the discrete solution at mesh interfaces, while the penalty parameters are chosen as complex numbers here instead of real numbers as usual. In this paper, it is proved that if the penalty parameter is a pure imaginary number i γ with 0 < γ ≤ C, then the CIP-FEM is stable (hence well posed) without any mesh constraint. Moreover, the error of the CIP-FEM in the H1 norm is bounded by C1kh + C2k3h2 when k 3h2 ≤ C0 and by C1kh + C 2/γ when k3h2>C0 and kh≤C3, where k is the wave number, h is the mesh size, and the C's are some positive constants independent of k, h and γ. By taking γ → 0+ in the CIP-FEM it is shown that if k3h2 ≤ C0, then the FEM attains a unique solution and satisfies the error estimate C1kh + C2k3h2 in the H 1 norm. Previous analytical results for the classical linear FEM in higher dimensions require the assumption that k2h is small, but it is too strict for a medium or high wave number. Optimal order L2-error estimates are also derived. Numerical results are provided to verify the theoretical findings and illustrate the performance of the two methods. In particular, it is shown that the pollution effect may be greatly reduced by tuning the penalty parameters in the CIP-FEM. The theoretical and numerical results reveal that the CIP-FEM has advantages over the FEM and is worthy of being investigated further. The pre-asymptotic error analysis of the hp version of the CIP-FEM and the FEM is studied in Part II. © 2013 The authors 2013. Published by Oxford University Press on behalf of the Institute of Mathematics and its Applications. All rights reserved.
We propose and study a domain decomposition method for solving the truncated perfectly matched layer (PML) approximation in bounded domain of Helmholtz scattering problems. The method is based on the decomposition of the domain into nonoverlapping layers and the idea of source transfer which transfers the sources equivalently layer by layer so that the solution in the final layer can be solved using a PML method defined locally outside the last two layers. The convergence of the method is proved for the case of constant wave number based on the analysis of the fundamental solution of the PML equation. The method can be used as an efficient preconditioner in the preconditioned GMRES method for solving discrete Helmholtz equations with constant and heterogeneous wave numbers. Numerical examples are included.
The uniaxial perfectly matched layer (PML) method uses a rectangular domain to define the PML problem and thus provides greater flexibility and efficiency in dealing with problems involving anisotropic scatterers. In this paper we first derive the uniaxial PML method for solving the time-domain scattering problem based on the Laplace transform and complex coordinate stretching in the frequency domain. We prove the long-time stability of the initial-boundary value problem of the uniaxial PML system for piecewise constant medium properties and show the exponential convergence of the time-domain uniaxial PML method. Our analysis shows that for fixed PML absorbing medium properties, any error of the time-domain PML method can be achieved by enlarging the thickness of the PML layer as ln T for large T > 0. Numerical experiments are included to illustrate the efficiency of the PML method.
We develop a stability and convergence theory for a Discontinuous Galerkin formulation (DG) of a highly indefinite Helmholtz problem in
R
d
,
d
∈
{
1
,
2
,
3
}
. The theory covers conforming as well as non-conforming generalized finite element methods. In contrast to conventional Galerkin methods where a minimal resolution condition is necessary to guarantee the unique solvability, it is proved that the DG-method admits a unique solution under much weaker conditions. As an application we present the error analysis for the
hp
hp
-version of the finite element method explicitly in terms of the mesh width
h
h
, polynomial degree
p
p
and wavenumber
k
k
. It is shown that the optimal convergence order estimate is obtained under the conditions that
kh
/
p
is sufficiently small and the polynomial degree
p
p
is at least
O
(
log
k
)
. On regular meshes, the first condition is improved to the requirement that
kh/p
kh
/
p
be sufficiently small.
We study the properties of a novel discontinuous Petrov Galerkin (DPG) method for acoustic wave propagation. The method yields Hermitian positive definite matrices and has good pre-asymptotic stability properties. Numerically, we find that the method exhibits negligible phase errors (otherwise known as pollution errors) even in the lowest order case. Theoretically, we are able to prove error estimates that explicitly show the dependencies with respect to the wavenumber ω, the mesh size h, and the polynomial degree p. But the current state of the theory does not fully explain the remarkably good numerical phase errors. Theoretically, comparisons are made with several other recent works that gave wave number explicit estimates. Numerically, comparisons are made with the standard finite element method and its recent modification for wave propagation with clever quadratures. The new DPG method is designed following the previously established principles of optimal test functions. In addition to the nonstandard test functions, in this work, we also use a nonstandard wave number dependent norm on both the test and trial space to obtain our error estimates.
A finite element method is developed for the numerical solution of scattering problems for the reduced wave equation. A new numerical approximation technique is presented, in which a finite dimensional space consisting of isoparametric trilinear elements defined on curved hexahedra, plus a finite element collection of spherical harmonics representing the scattered field, is utilized. In addition, the stiffness matrices are positive definite, and Gaussian elimination without pivoting can be used.
A cognitive journey towards the reliable simulation of scattering problems using finite element methods, with the pre-asymptotic analysis of Galerkin FEM for the Helmholtz equation with moderate and large wave number forming the core of this book. Starting from the basic physical assumptions, the author methodically develops both the strong and weak forms of the governing equations, while the main chapter on finite element analysis is preceded by a systematic treatment of Galerkin methods for indefinite sesquilinear forms. In the final chapter, three dimensional computational simulations are presented and compared with experimental data. The author also includes broad reference material on numerical methods for the Helmholtz equation in unbounded domains, including Dirichlet-to-Neumann methods, absorbing boundary conditions, infinite elements and the perfectly matched layer. A self-contained and easily readable work.
In this paper, which is part II in a series of two, the investigation of the Galerkin finite element solution to the Helmholtz equation is continued. While part I contained results on the h version with piecewise linear approximation, the present part deals with approximation spaces of order . As in part I, the results are presented on a one-dimensional model problem with Dirichlet--Robin boundary conditions. In particular, there are proven stability estimates, both with respect to data of higher regularity and data that is bounded in lower norms. The estimates are shown both for the continuous and the discrete spaces under consideration. Further, there is proven a result on the phase difference between the exact and the Galerkin finite element solutions for arbitrary p that had been previously conjectured from numerical experiments. These results and further preparatory statements are then employed to show error estimates for the Galerkin finite element method (FEM). It becomes evident that the error estimate for higher approximation can---with certain assumptions on the data---be written in the same form as the piecewise linear case, namely, as the sum of the error of best approximation plus a pollution term that is of the order of the phase difference. The paper is concluded with a numerical evaluation.
We describe a practical procedure for the triangulation of arbitrary n-dimensional regular bounded domains by isoparametric simplicial elements preserving the optimal order of accuracy. Along similar lines we prove general error estimates for the finite element solution of second order elliptic problems. Various kinds of nonhomogeneous boundary conditions can be dealt with, such as Dirichlet, Neumann and Robin boundary conditions.
This paper is devoted to a general theory of approximation of functions in finite-element spaces. In particular, the case is considered where the functions to be interpolated are on the one hand not very smooth, and on the other are defined on curved domains. Thus, a number of well-known optimal interpolation results are generalized.
In this paper we consider a class of robust multilevel precontioners for the
Helmholtz equation with high wave number. The key idea in this work is to use
the continuous interior penalty finite element methods (CIP-FEM) studied in
\cite{Wu12,Wu12-hp} to construct the stable coarse grid correction problems.
The multilevel methods, based on GMRES smoothing on coarse grids, are then
served as a preconditioner in the outer GMRES iteration. In the one dimensional
case, convergence property of the modified multilevel methods is analyzed by
the local Fourier analysis. From our numerical results, we find that the
proposed methods are efficient for a reasonable range of frequencies. The
performance of the algorithms depends relatively mildly on wave number. In
particular, only one GMRES smoothing step may guarantee the optimal convergence
of our multilevel algorithm, which remedies the shortcoming of the multilevel
algorithm in \cite{EEO01}.
This paper addresses the properties of Continuous Interior Penalty (CIP)
finite element solutions for the Helmholtz equation. The h-version of the CIP
finite element method with piecewise linear approximation is applied to a
one-dimensional model problem. We first show discrete well posedness and
convergence results, using the imaginary part of the stabilization operator,
for the complex Helmholtz equation. Then we consider a method with real valued
penalty parameter and prove an error estimate of the discrete solution in the
-norm, as the sum of best approximation plus a pollution term that is the
order of the phase difference. It is proved that the pollution can be
eliminated by selecting the penalty parameter appropriately. As a result of
this analysis, thorough and rigorous understanding of the error behavior
throughout the range of convergence is gained. Numerical results are presented
that show sharpness of the error estimates and highlight some phenomena of the
discrete solution behavior.
This paper analyzes the error estimates of the hybridizable discontinuous
Galerkin (HDG) method for the Helmholtz equation with high wave number in two
and three dimensions. The approximation piecewise polynomial spaces we deal
with are of order . Through choosing a specific parameter and using
the duality argument, it is proved that the HDG method is stable without any
mesh constraint for any wave number . By exploiting the stability
estimates, the dependence of convergence of the HDG method on and
p is obtained. Numerical experiments are given to verify the theoretical
results.
A rigorous convergence theory for Galerkin methods for a model Helmholtz problem in R d , d ∈ {1, 2, 3} is presented. General conditions on the approximation properties of the approximation space are stated that ensure quasi-optimality of the method. As an application of the general theory, a full error analysis of the classical hp-version of the finite element method (hp-FEM) is presented for the model problem where the dependence on the mesh width h, the approximation order p, and the wave number k is given explicitly. In particular, it is shown that quasi-optimality is obtained under the conditions that kh/p is sufficiently small and the polynomial degree p is at least O(log k).
The Bérenger perfectly matched layer (PML) is used in computational electromagnetism as a “sponge layer” to terminate finite element approximations of scattering problems. An infinite perfectly matched layer creates no reflection for incident waves of any frequency or any incident direction, and the waves decay exponentially in magnitude into the layer. This property is lost when the layer is truncated since the truncation boundary generates a reflected wave. Discretization of the differential equations further changes the reflection coefficient of the layer. In this talk we describe the PML in the simple case of a two dimensional electromagnetic problem. The first part of this abstract is expository. We show how to derive the layer using a change of variable approach. After this derivation we describe some new analysis to show the effects of truncating and discretizing the PML F. Collino and P. Monk [cf. the authors, SIAM J. Sci. Comput. 19, 2061-2090 (1998; Zbl 0940.78011)]. We end by showing how to compute an optimal layer, and make some remarks about PML for the full Maxwell system.
The paper addresses the properties of finite element solutions for the Helmholtz equation. The h-version of the finite element method with piecewise linear approximation is applied to a one-dimensional model problem. New results are shown on stability and error estimation of the discrete model. In all propositions, assumptions are made on the magnitude of hk only, where k is the wavelength and h is the stepwidth of the FE-mesh. Previous analytical results had been shown with the assumption that k2h is small. For medium and high wavenumber, these results do not cover the meshsizes that are applied in practical applications. The main estimate reveals that the error in H1-norm of discrete solutions for the Helmholtz equation is polluted when k2h is not small. The error is then not quasioptimal; i.e., the relation of the FE-error to the error of best approximation generally depends on the wavenumber k. It is noted that the pollution term in the relative error is of the same order as the phase lead of the numerical solution. In the result of this analysis, thorough and rigorous understanding of error behavior throughout the range of convergence is gained. Numerical results are presented that show sharpness of the error estimates and highlight some phenomena of the discrete solution behavior. The h-p-version of the FEM is studied in Part II.
We consider the approximation of the frequency domain three-dimensional Maxwell scattering problem using a truncated domain perfectly matched layer (PML). We also treat the time-harmonic PML approximation to the acoustic scattering problem. Following work of Lassas and Somersalo in 1998, a transitional layer based on spherical geometry is defined, which results in a constant coefficient problem outside the transition. A truncated (computational) domain is then defined, which covers the transition region. The truncated domain need only have a minimally smooth outer boundary (e.g., Lipschitz continuous). We consider the truncated PML problem which results when a perfectly conducting boundary condition is imposed on the outer boundary of the truncated domain. The existence and uniqueness of solutions to the truncated PML problem will be shown provided that the truncated domain is sufficiently large, e.g., contains a sphere of radius R_t . We also show exponential (in the parameter R_t ) convergence of the truncated PML solution to the solution of the original scattering problem inside the transition layer. Our results are important in that they are the first to show that the truncated PML problem can be posed on a domain with nonsmooth outer boundary. This allows the use of approximation based on polygonal meshes. In addition, even though the transition coefficients depend on spherical geometry, they can be made arbitrarily smooth and hence the resulting problems are amenable to numerical quadrature. Approximation schemes based on our analysis are the focus of future research.
We develop a stability and convergence theory for a class of highly indefinite elliptic boundary value problems (bvps) by considering the Helmholtz equation at high wavenumber k as our model problem. The key element in this theory is a novel k-explicit regularity theory for Helmholtz bvps that is based on decomposing the solution into two parts: the first part has the Sobolev regularity properties expected of second order elliptic PDEs but features k-independent regularity constants; the second part is an analytic function for which k-explicit bounds for all derivatives are given. This decomposition is worked out in detail for several types of bvps, namely, the Helmholtz equation in bounded smooth domains or convex polygonal domains with Robin boundary conditions and in exterior domains with Dirichlet boundary conditions. We present an error analysis for the classical hp-version of the finite element method (hp-FEM) where the dependence on the mesh width h, the approximation order p, and the wavenumber k is given explicitly. In particular, under the assumption that the solution operator for Helmholtz problems is polynomially bounded in k, it is shown that quasi optimality is obtained under the conditions that kh/p is sufficiently small and the polynomial degree p is at least O(log k).
We develop an adaptive perfectly matched layer (PML) technique for solving the time harmonic scattering problems. The PML parameters such as the thickness of the layer and the ctitious medium property are determined through sharp a posteriori error estimates. The derived nite element a posteriori estimate for adapting meshes has the nice feature that it decays exponentially away from the boundary of the xed domain where the PML layer is placed. This property makes the total computational costs insensitive to the thickness of the PML absorbing layers. Numerical experiments are included to illustrate the competitive behavior of the proposed adaptive method.
We develop a finite element adaptive strategy with error control for the wave scattering by periodic structures. The unbounded computational domain is truncated to a bounded one by an extension of the perfectly matched layer (PML) technique, which attenuates both the outgoing and evanescent waves in the PML region. PML parameters such as the thickness of the layer and the medium property are determined through sharp a posteriori error estimates. Numerical experiments are included to illustrate the competitive behavior of the proposed adaptive method.
The perfectly matched layer is a technique of free-space simulation developed for solving unbounded electromagnetic problems with the finite-difference time-domain method. Referred to as PML, this technique has been described in a previous paper for two-dimensional problems. The present paper is devoted to the three-dimensional case.The theory of the perfectly matched layer is generalized to three dimensions and some numerical experiments are shown to illustrate the efficiency of this new method of free-space simulation. 20 refs., 11 figs.
Thesis (Ph. D.)--University of Maryland at College Park, 1995. Thesis research directed by Dept. of Mathematics. Includes bibliographical references (leaves 224-227).
A modified set of Maxwell's equations is presented that includes complex coordinate stretching along the three cartesian coordinates. The added degrees of freedom in the modified Maxwell's equations allow the specification of absorbing boundaries with zero reflection at all angles of incidence and all frequencies. The modified equations are also related to the perfectly matched layer that was presented recently for 2-D wave propagation. Absorbing material boundary conditions are of particular interest for finite difference time domain (FDTD) computations on a single-instruction multiple-data (SIMD) massively parallel supercomputer. A 3-D FDTD algorithm has been developed on a Connection Machine CM-5 based on the modified Maxwell's equations and simulation results are presented to validate the approach.
By complex coordinate stretching and a change of variables, it is shown simply that PML is reflectionless for all frequencies and all angles. Also, Maxwell's equations for PML media reduce to ordinary Maxwell's equations with complex coordinate systems. Many closed form solutions for Maxwell's equations map to corresponding closed form solutions in complex coordinate systems. Numerical simulations with the closed form solutions show that metallic boxes lined with PML media are highly absorptive. These closed form solutions lend a better understanding to the absorptive properties of PML media. For instance, they explain why a PML medium is absorptive when a dielectric or metallic interface extends to the edge to a simulation region where PML media reside. More importantly, the complex coordinate stretching method can be generalized to non-Cartesian coordinate systems, providing absorbing boundary conditions in these coordinate systems. 1. Introduction The perfectly matched layer (PML) ...
- J Douglas
- T Dupont
J. Douglas and T. Dupont, Interior Penalty Procedures for Elliptic and Parabolic Galerkin
Methods, Lecture Notes in Physics, 58 (1976), pp. 207-216.
- X Feng
- H Wu
X. Feng and H. Wu, hp-discontinuous Galerkin methods for the Helmholtz equation with
large wave number, Math. Comp., 80 (2011), pp. 1997-2024.