Error analysis of an enhanced DtN-FE method for exterior scattering problems

In this work we analyze the convergence of the high-order Enhanced DtN-FEM algorithm, described in our previous work (Nicholls and Nigam, J. Comput. Phys. 194:278–303, 2004), for solving exterior acoustic scattering problems in . This algorithm consists of using an exact Dirichlet-to-Neumann (DtN) map on a hypersurface enclosing the scatterer, where the hypersurface is a perturbation of a circle, and, in practice, the perturbation can be very large. Our theoretical work had shown the DtN map was analytic as a function of this perturbation. In the present work, we carefully analyze the error introduced by virtue of using this algorithm. Specifically, we give a full account of the error introduced by truncating the DtN map at a finite order in the perturbation expansion, and study the well-posedness of the associated formulation. During computation, the Fourier series of the Dirichlet data on the artificial boundary must be truncated. To deal with the ensuing loss of uniqueness of solutions, we propose a modified DtN map, and prove well-posedness of the resulting problem. We quantify the spectral error introduced due to this truncation of the data. The key tools in the analysis include a new theorem on the analyticity of the DtN map in a suitable Sobolev space, and another on the perturbation of non-self-adjoint Fredholm operators.     

Error analysis of the DtN-FEM for the scattering problem in acoustics via Fourier analysis

In this paper, we are concerned with the error analysis for the finite element solution of the two-dimensional exterior Neumann boundary value problem in acoustics. In particular, we establish explicit priori error estimates in H¹ and L²- norms including both the effect of the truncation of the DtN mapping and that of the numerical discretization. To apply the finite element method (FEM) to the exterior problem, the original boundary value problem is reduced to an equivalent nonlocal boundary value problem via a Dirichlet-to-Neumann (DtN) mapping represented in terms of the Fourier expansion series. We discuss essential features of the corresponding variational equation and its modification due to the truncation of the DtN mapping in appropriate function spaces. Numerical tests are presented to validate our theoretical results.     

Three-dimensional approximate local DtN boundary conditions for prolate spheroid boundaries

We propose a new class of approximate local DtN boundary conditions to be applied on prolate spheroidal-shaped exterior boundaries when solving problems of acoustic scattering by elongated obstacles. These conditions are: (a) exact for the first modes, (b) easy to implement and to parallelize, (c) compatible with the local structure of the computational finite element scheme, and (d) applicable to exterior ellipsoidal-shaped boundaries that are more suitable in terms of cost-effectiveness for surrounding elongated scatterers. We investigate analytically and numerically the effect of the frequency regime and the slenderness of the boundary on the accuracy of these conditions. We also compare their performance to the second-order absorbing boundary condition (BGT2) designed by Bayliss, Gunzburger and Turkel when expressed in prolate spheroid coordinates. The analysis reveals that, in the low-frequency regime, the new second-order DtN condition (DtN2) retains a good level of accuracy regardless of the slenderness of the boundary. In addition, the DtN2 boundary condition outperforms the BGT2 condition. Such superiority is clearly noticeable for large eccentricity values.     

Inverse boundary value problem for ocean acoustics

For an ocean with constant depth and rigid bottom which contains compactly supported inhomogeneity of the water sound velocity, we prove uniqueness for the identification of the inhomogeneity from the Dirichlet‐to‐Neumann (DtN) map on the surface of a bounded region containing the inhomogeneity. The DtN map is the map which maps the pressure applied on the boundary of this region to the corresponding flux (displacement). In an analogous geometric configuration and with similar boundary conditions, the uniqueness for the inverse electroconductivity problem from the DtN map (i.e. voltage‐to‐current map) can be proved in the same framework. Copyright © 2001 John Wiley & Sons, Ltd.     

Error estimates of the finite element method for the exterior Helmholtz problem with a modified DtN boundary condition

A priori error estimates in the H¹- and L²-norms are established for the finite element method applied to the exterior Helmholtz problem, with modified Dirichlet-to-Neumann (MDtN) boundary condition. The error estimates include the effect of truncation of the MDtN boundary condition as well as that of discretization of the finite element method. The error estimate in the L²-norm is sharper than that obtained by the author [D. Koyama, Error estimates of the DtN finite element method for the exterior Helmholtz problem, J. Comput. Appl. Math. 200 (1) (2007) 21-31] for the truncated DtN boundary condition.     

Error estimates of the DtN finite element method for the exterior Helmholtz problem

A priori error estimates are established for the DtN (Dirichlet-to-Neumann) finite element method applied to the exterior Helmholtz problem. The error estimates include the effect of truncation of the DtN boundary condition as well as that of the finite element discretization. A property of the Hankel functions which plays an important role in the proof of the error estimates is introduced.     

A wavelets method for plane elasticity problem

A Neumann boundary value problem of plane elasticity problem in the exterior circular domain is reduced into an equivalent natural boundary integral equation and a Poisson integral formula with the DtN method. Using the trigonometric wavelets and Galerkin method, we obtain a fast numerical method for the natural boundary integral equation which has an unique solution in the quotient space. We decompose the stiffness matrix in our numerical method into four circulant and symmetrical or antisymmetrical submatrices, and hence the solution of the associated linear algebraic system can be solved with the fast Fourier transform (FFT) and the inverse fast Fourier transform (IFFT) instead of the inverse matrix. Examples are given for demonstrating our method has good accuracy of our method even though the exact solution is almost singular.     

Analysis of a coupled finite-infinite element method for exterior Helmholtz problems

Summary. This analysis of convergence of a coupled FEM-IEM is based on our previous work on the FEM and the IEM for exterior Helmholtz problems. The key idea is to represent both the exact and the numerical solution by the Dirichlet-to-Neumann operators that they induce on the coupling hypersurface in the exterior of an obstacle. The investigation of convergence can then be related to a spectral analysis of these DtN operators. We give a general outline of our method and then proceed to a detailed investigation of the case that the coupling surface is a sphere. Our main goal is to explore the convergence mechanism. In this context, we show well-posedness of both the continuous and the discrete models. We further show that the discrete inf-sup constants have a positive lower bound that does not depend on the number of DOF of the IEM. The proofs are based on lemmas on the spectra of the continuous and the discrete DtN operators, where the spectral characterization of the discrete DtN operator is given as a conjecture from numerical experiments. In our convergence analysis, we show algebraic (in terms of N) convergence of arbitrary order and generalize this result to exponential convergence. Received April 10, 1999 / Revised version received November 10, 1999 / Published online October 16, 2000     

Solution of exterior problem using ellipsoidal artificial boundary

In this paper, we present a general ellipsoidal artificial boundary method for three-dimensional exterior problem. The exact artificial boundary condition, which is expressed explicitly by the series concerning the ellipsoidal harmonic functions, is derived and then an equivalent problem in a bounded domain is presented. The error estimates show that the convergence rate depends on the mesh parameter, the number of terms used in the exact artificial boundary condition, and the location of the artificial boundary.     

Stability analysis of FDTD to UPML for time dependent Maxwell equations

We study an finite-difference time-domain (FDTD) system of uniaxial perfectly matched layer (UPML) method for electromagnetic scattering problems. Particularly we analyze the discrete initial-boundary value problems of the transverse magnetic mode (TM) to Maxwell’s equations with Yee’s algorithm. An exterior domain in two spacial dimension is truncated by a square with a perfectly matched layer filled by a certain artificial medium. Besides, an artificial boundary condition is imposed on the outer boundary of the UPML. Using energy method, we obtain the stability of this FDTD system on the truncated domain. Numerical experiments are designed to approve the theoretical analysis.      

Dirichlet-Neumann alternating algorithm for an exterior anisotropic quasilinear elliptic problem

In this paper, by the Kirchhoff transformation, a Dirichlet-Neumann (D-N) alternating algorithm which is a non-overlapping domain decomposition method based on natural boundary reduction is discussed for solving exterior anisotropic quasilinear problems with circular artificial boundary. By the principle of the natural boundary reduction, we obtain natural integral equation for the anisotropic quasilinear problems on circular artificial boundaries and construct the algorithm and analyze its convergence. Moreover, the convergence rate is obtained in detail for a typical domain. Finally, some numerical examples are presented to illustrate the feasibility of the method.     

A FEM–DtN formulation for a non‐linear exterior problem in incompressible elasticity

In this paper, we combine the usual finite element method with a Dirichlet‐to‐Neumann (DtN) mapping, derived in terms of an infinite Fourier series, to study the solvability and Galerkin approximations of an exterior transmission problem arising in non‐linear incompressible 2d‐elasticity. We show that the variational formulation can be written in a Stokes‐type mixed form with a linear constraint and a non‐linear main operator. Then, we provide the uniqueness of solution for the continuous and discrete formulations, and derive a Cea‐type estimate for the associated error. In particular, our error analysis considers the practical case in which the DtN mapping is approximated by the corresponding finite Fourier series. Finally, a reliable a posteriori error estimate, well suited for adaptive computations, is also given. Copyright © 2003 John Wiley & Sons, Ltd.     

A numerical method for problems in infinite strips with irregularities extending to infinity

The Dirichlet-to-Neumann (DtN) Finite Element Method is a combined analytic-numerical method for boundary value problems in infinite domains. The use of this method is usually based on the assumption that, in the infinite domain D exterior to the finite computational domain, the governing differential equations are sufficiently simple. In particular, in D it is generally assumed that the equations are linear, homogeneous, and have constant coefficients. In this article, an extension of the DtN method is proposed, which can be applied to elliptic problems with “irregularities” in the exterior domain D, such as (a) inhomogeneities, (b) variable coefficients, and (c) nonlinearities. This method is based on iterative “regularization” of the problem in D, and on the efficient treatment of infinite-domain integrals. Semi-infinite strip problems are used for illustrating the method. Convergence of the iterative process is analyzed both theoretically and numerically. Nonuniformity difficulties and a way to overcome them are discussed. © 1998 John Wiley & Sons, Inc. Numer Methods Partial Differential Eq 14:233–249, 1998     

DOMAIN DECOMPOSITION WITH NONMATCHING GRIDS FOR EXTERIOR TRANSMISSION PROBLEMS VIA FEM AND DTN MAPPING

In this paper, we are concerned with a non-overlapping domain decomposition method （DDM） for exterior transmission problems in the plane. Based on the natural boundary integral operator, we combine the DDM with a Dirichlet-to-Neumann （DtN） mapping and provide the numerical analysis with nonmatching grids. The weak continuity of the approximation solutions on the interface is imposed by a dual basis multiplier. We show that this multiplier space can generate optimal error estimate and obtain the corresponding rate of convergence. Finally, several numerical examples confirm the theoretical results.     

Corrections to Lee's visibility polygon algorithm

We present a modification and extension of the (linear time) visibility polygon algorithm of Lee. The algorithm computes the visibility polygon of a simple polygon from a viewpoint that is either interior to the polygon, or in its blocked exterior (the cases of viewpoints on the boundary or in the free exterior being simple extensions of the interior case). We show by example that the original algorithm by Lee, and a more complex algorithm by El Gindy and Avis, can fail for polygons that wind sufficiently. We present a second version of the algorithm, which does not extend to the blocked exterior case.This work was partially supported by grants from the Central Research Fund of the University of Alberta and the Natural Sciences and Engineering Research Council of Canada.     

Stability of a finite element method for 3D exterior stationary Navier-Stokes flows

We consider numerical approximations of stationary incompressible Navier-Stokes flows in 3D exterior domains, with nonzero velocity at infinity. It is shown that a P1-P1 stabilized finite element method proposed by C. Rebollo: A term by term stabilization algorithm for finite element solution of incompressible flow problems, Numer. Math. 79 (1998), 283–319, is stable when applied to a Navier-Stokes flow in a truncated exterior domain with a pointwise boundary condition on the artificial boundary.     

A domain decomposition method for linear exterior boundary value problems

In this paper, we present a domain decomposition method, based on the general theory of Steklov-Poincaré operators, for a class of linear exterior boundary value problems arising in potential theory and heat conductivity. We first use a Dirichlet-to-Neumann mapping, derived from boundary integral equation methods, to transform the exterior problem into an equivalent mixed boundary value problem on a bounded domain. This domain is decomposed into a finite number of annular subregions, and the Dirichlet data on the interfaces is introduced as the unknown of the associated Steklov-Poincaré problem. This problem is solved with the Richardson method by introducing a Dirichlet-Robin-type preconditioner, which yields an iteration-by-subdomains algorithm well suited for parallel computations. The corresponding analysis for the finite element approximations and some numerical experiments are also provided.     

A stream function-vorticity formulation coupled with boundary integrals for the two-dimensional exterior Stokes problem

In this work, we derive a stream function-vorticity variational formulation coupled with boundary integrals for the exterior Stokes problem in two dimensions, when the right-hand side has a bounded support. The stream function-vorticity formulation is expressed in a bounded region containing the support of the right-hand side, and the boundary conditions on the artificial boundary are obtained by an integral representation. We prove that this coupled formulation is equivalent to the original Stokes problem.     

Exterior electromagnetic shaping using wavelet BEM

The present paper is devoted to exterior electromagnetic shaping in two dimensions. We model the conductors by regular densities which leads to a finite objective and allows a line‐search. In order to compute the surface pressure we optimize an Augmented Lagrangian by a Newton method using a second‐order approach for the Lagrange multiplier. Since the underlying state function satisfies an exterior boundary value problem, we compute first and second order derivatives of its boundary data by boundary integral equations which are numerically solved by a fast wavelet Galerkin scheme. Numerical results prove that we succeeded in finding a fast and robust algorithm for solving the considered class of problems. Copyright © 2004 John Wiley & Sons, Ltd.     

A finite-element capacitance matrix method for exterior Helmholtz problems

Summary. We introduce an algorithm for the efficient numerical solution of exterior boundary value problems for the Helmholtz equation. The problem is reformulated as an equivalent one on a bounded domain using an exact non-local boundary condition on a circular artificial boundary. An FFT-based fast Helmholtz solver is then derived for a finite-element discretization on an annular domain. The exterior problem for domains of general shape are treated using an imbedding or capacitance matrix method. The imbedding is achieved in such a way that the resulting capacitance matrix has a favorable spectral distribution leading to mesh independent convergence rates when Krylov subspace methods are used to solve the capacitance matrix equation. Received May 2, 1995