Natural Boundary Integral Method and its New Development

In this paper, the natural boundary integral method, and some related methods, including coupling method of the natural boundary elements and finite elements, which is also called DtN method or the method with exact artificial boundary conditions, domain decomposition methods based on the natural boundary reduction, and the adaptive boundary element method with hyper-singular a posteriori error estimates, are discussed.     

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.     

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.     

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 analysis and preconditioning for an enhanced DtN-FE algorithm for exterior scattering problems

In this paper we present an error analysis for a high-order accurate combined Dirichlet-to-Neumann (DtN) map/finite element (FE) algorithm for solving two-dimensional exterior scattering problems. We advocate the use of an exact DtN (or Steklov–Poincaré) map at an artificial boundary exterior to the scatterer to truncate the unbounded computational region. The advantage of using an exact DtN map is that it provides a transparent condition which does not reflect scattered waves unphysically. Our algorithm allows for the specification of quite general artificial boundaries which are perturbations of a circle. To compute the DtN map on such a geometry we utilize a boundary perturbation method based upon recent theoretical work concerning the analyticity of the DtN map. We also present some preliminary work concerning the preconditioning of the resulting system of linear equations, including numerical experiments.     

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.     

A coupled BEM and FEM for the interior transmission problem in acoustics

The interior transmission problem (ITP) is a boundary value problem arising in inverse scattering theory, and it has important applications in qualitative methods. In this paper, we propose a coupled boundary element method (BEM) and a finite element method (FEM) for the ITP in two dimensions. The coupling procedure is realized by applying the direct boundary integral equation method to define the so-called Dirichlet-to-Neumann (DtN) mappings. We show the existence of the solution to the ITP for the anisotropic medium. Numerical results are provided to illustrate the accuracy of the coupling 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.     

RECOVERY BASED FINITE ELEMENT METHOD FOR BIHARMONIC EQUATION IN 2D

We design and numerically validate a recovery based linear finite element method for solving the biharmonic equation.The main idea is to replace the gradient operator▽on linear finite element space by G(▽)in the weak formulation of the biharmonic equation,where G is the recovery operator which recovers the piecewise constant function into the linear finite element space.By operator G,Laplace operator△is replaced by▽·G(▽).Furthermore,the boundary condition on normal derivative▽u-n is treated by the boundary penalty method.The explicit matrix expression of the proposed method is also introduced.Numerical examples on the uniform and adaptive meshes are presented to illustrate the correctness and effectiveness of the proposed method.     

Well-Posedness and Finite Element Approximations for Elliptic SPDEs with Gaussian Noises

The paper studies the well-posedness and optimal error estimates of spectral finite element approximations for the boundary value problems of semi-linear elliptic SPDEs driven by white or colored Gaussian noises. The noise term is approximated through the spectral projection of the covariance operator, which is not required to be commutative with the Laplacian operator.Through the convergence analysis of SPDEs with the noise terms replaced by the projected noises, the well-posedness of the SPDE is established under certain covariance operator-dependent conditions. These SPDEs with projected noises are then numerically approximated with the finite element method. A general error estimate framework is established for the finite element approximations. Based on this framework, optimal error estimates of finite element approximations for elliptic SPDEs driven by power-law noises are obtained. It is shown that with the proposed approach, convergence order of white noise driven SPDEs is improved by half for one-dimensional problems, and by an infinitesimal factor for higher-dimensional problems.     

Study of the initial value problems appearing in a method of factorization of second-order elliptic boundary value problems

In [J. Henry, A.M. Ramos, Factorization of second order elliptic boundary value problems by dynamic programming, Nonlinear Analysis. Theory, Methods & Applications 59 (2004) 629–647] we presented a method for factorizing a second-order boundary value problem into a system of uncoupled first-order initial value problems, together with a nonlinear Riccati type equation for functional operators. A weak sense was given to that system but we did not perform a direct study of those equations. This factorization utilizes either the Neumann to Dirichlet (NtD) operator or the Dirichlet to Neumann (DtN) operator, which satisfy a Riccati equation. Here we consider the framework of Hilbert–Schmidt operators, which provides tools for a direct study of this Riccati type equation. Once we have solved the system of Cauchy problems, we show that its solution solves the original second-order boundary value problem. Finally, we indicate how this techniques can be used to find suitable transparent conditions.     

A Symmetric Galerkin Boundary Element Method for Linear Poroelasticity

In linear poroelasticity so far only collocation boundary element methods have been available. However, in some applications, e.g., when coupling with finite elements is desired, a symmetric formulation is preferable. Choosing a Galerkin approach which involves the second boundary integral equation, such a formulation is possible. Here, a previously presented integration by part technique for the regularization of the first boundary integral equation is extended to the second boundary integral equation as well. While the weakly singular representation of the double layer operator has been presented before, the emphasis lies here on the so called hyper-singular boundary integral operator. Due to the regularization, this operator can be evaluated numerically and, hence, be used within a numerical scheme for the first time. Different numerical studies will be presented to show the behavior of the established symmetric Galerkin boundary element method, also comparing it with collocation boundary element methods. (© 2011 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

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.     

Implicit residual error estimators for the coupling of finite elements and boundary elements

We consider a symmetric Galerkin method for the coupling of finite elements and boundary elements for elliptic problems with a monotone operator in the finite element domain. We derive an a posteriori error estimator which involves the solution of equilibrated local Neumann problems in the finite element domain and requires computation of a residual term on the coupling interface. Finally, we discuss a similar approach for a coupling with Signorini contact conditions on the interface. Copyright © 1999 John Wiley & Sons, Ltd.     

CASCADIC MULTIGRID METHODS FOR MORTAR WILSON FINITE ELEMENT METHODS ON PLANAR LINEAR ELASTICITY

Cascadic multigrid technique for mortar Wilson finite element method of homogeneous boundary value planar linear elasticity is described and analyzed. First the mortar Wilson finite element method for planar linear elasticity will be analyzed, and the error estimate under L2 and H1 norm is optimal. Then a cascadic multigrid method for the mortar finite element discrete problem is described. Suitable grid transfer operator and smoother are developed which lead to an optimal cascadic multigrid method. Finally, the computational results are presented.     

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.     

On the symmetric boundary element method and the symmetric coupling of boundary elements and finite elements

The finite element method and the boundary element method areamong the most frequently applied tools in the numerical treatmentof partial differential equations. However, their propertiesappear to be complementary: while the boundary element methodis appropriate for the most important linear partial differentialequations with constant coefficients in bounded or unboundeddomains, the finite element method seems to be more appropriatefor inhomogeneous or even nonlinear problems. but is somehowrestricted to bounded domains. The symmetric coupling of thetwo methods inherits the advantages of both methods. This paper treats the symmetric coupling of finite elementsand boundary elements for a model transmission problem in twoand three dimensions where we have two domains: a bounded domainwith nonlinear, even plastic material behaviour, is surroundedby an unbounded, exterior, domain with isotropic homogeneouslinear elastic material. Practically. the coupling is performedsuch that the boundary element method contributes a macro-element,like a large finite element, within a standard finite elementanalysis program. Emphasis is on two-dimensional problems wherethe approach using the Poincaré-Steklov operator seemsto be impossible at first glance. E-mail: cc{at}numerik.uni-kiel.de     

Convergence of a multiscale finite element method for elliptic problems with rapidly oscillating coefficients

We propose a multiscale finite element method for solving second order elliptic equations with rapidly oscillating coefficients. The main purpose is to design a numerical method which is capable of correctly capturing the large scale components of the solution on a coarse grid without accurately resolving all the small scale features in the solution. This is accomplished by incorporating the local microstructures of the differential operator into the finite element base functions. As a consequence, the base functions are adapted to the local properties of the differential operator. In this paper, we provide a detailed convergence analysis of our method under the assumption that the oscillating coefficient is of two scales and is periodic in the fast scale. While such a simplifying assumption is not required by our method, it allows us to use homogenization theory to obtain a useful asymptotic solution structure. The issue of boundary conditions for the base functions will be discussed. Our numerical experiments demonstrate convincingly that our multiscale method indeed converges to the correct solution, independently of the small scale in the homogenization limit. Application of our method to problems with continuous scales is also considered.

     

On the Maxwell system under impedance boundary conditions with memory

We consider an initial boundary value problem for the system of the Maxwell equations in a bounded domain with smooth boundary on a finite time interval with new boundary conditions with memory. In appropriate function spaces, we define and study the nonselfadjoint operator that is generated by the Maxwell operator under a boundary condition with memory. Using the operator method, we prove an existence and uniqueness theorem for a solution to the initial boundary value problem.     

Three-dimensional simulation of flat rolling through a combined finite element–boundary element approach

This paper presents an innovative approach for analysing three-dimensional flat rolling. The proposed approach is based on a solution resulting from the combination of the finite element method with the boundary element method. The finite element method is used to perform the rigid–plastic numerical modelling of the workpiece allowing the estimation of the roll separating force, rolling torque and contact pressure along the surface of the rolls. The boundary element method is applied for computing the elastic deformation of the rolls. The combination of the two numerical methods is made using the finite element solution of the contact pressure along the surface of the rolls to define the boundary conditions to be applied on the elastic analysis of the rolls. The validity of the proposed approach is discussed by comparing the theoretical predictions with experimental data found in the literature.