An Inverse Transmission Scattering Problem and the Enclosure Method

We consider an inverse scattering problem in two-dimensions for a penetrable polygonal obstacle having different density from the back ground medium, however, the speed of sound is constant in the whole space. Using a single set of the Cauchy data of the response for a single incident plane wave with a fixed wave number on a circle surrounding the obstacle, we give an extraction formula of the convex hull of the obstacle. An algorithm based on the formula is also described.     

h-p Spectral Element Method for Elliptic Problems on Non-smooth Domains Using Parallel Computers

We propose a new h-p spectral element method to solve elliptic boundary value problems with mixed Neumann and Dirichlet boundary conditions on non-smooth domains. The method is shown to be exponentially accurate and asymptotically faster than the standard h-p finite element method. The spectral element functions are fully non-conforming for pure Dirichlet problems and conforming only at the vertices of the elements for mixed problems, and hence, the dimension of the resulting Schur complement matrix is quite small. The method is a least-squares collocation method and the resulting normal equations are solved using preconditioned conjugate gradient method with an almost optimal preconditioner. The algorithm is suitable for a distributed memory parallel computer. The numerical results of a number of model problems are presented, which confirm the theoretical estimates.     

On the existence and convergence of the solution of PML equations

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. Partly supported by the Finnish Academy, project 37692.     

A Numerical Method to Verify the Invertibility of Linear Elliptic Operators with Applications to Nonlinear Problems

In this paper, we propose a numerical method to verify the invertibility of second-order linear elliptic operators. By using the projection and the constructive a priori error estimates, the invertibility condition is formulated as a numerical inequality based upon the existing verification method originally developed by one of the authors. As a useful application of the result, we present a new verification method of solutions for nonlinear elliptic problems, which enables us to simplify the verification process. Several numerical examples that confirm the actual effectiveness of the method are presented.     

Domain decomposition methods for the neutron diffusion problem

The neutronic simulation of a nuclear reactor core is performed using the neutron transport equation, and leads to an eigenvalue problem in the steady-state case. Among the deterministic resolution methods, simplified transport (S_PN$S{P}_N$) or diffusion approximations are often used. The MINOS solver developed at CEA Saclay uses a mixed dual finite element method for the resolution of these problems, and has shown his efficiency. In order to take into account the heterogeneities of the geometry, a very fine mesh is generally required, and leads to expensive calculations for industrial applications. In order to take advantage of parallel computers, and to reduce the computing time and the local memory requirement, we propose here two domain decomposition methods based on the MINOS solver. The first approach is a component mode synthesis method on overlapping subdomains: several eigenmodes solutions of a local problem on each subdomain are taken as basis functions used for the resolution of the global problem on the whole domain. The second approach is an iterative method based on a non-overlapping domain decomposition with Robin interface conditions. At each iteration, we solve the problem on each subdomain with the interface conditions given by the solutions on the adjacent subdomains estimated at the previous iteration. Numerical results on parallel computers are presented for the diffusion model on realistic 2D and 3D cores.     

A variational approximation method for 2nd order elliptic eigenvalue problems in a composite structure with nonstandard boundary conditions

In this paper we deal with the finite element analysis of a class of eigenvalue problems (EVPs) in a composite structure in the plane, consisting of rectangular subdomains which enclose an intermediate region. Nonlocal boundary conditions (BCs) of Robin type are imposed on the inner boundaries, i.e. on the interfaces of the respective subdomains with the intermediate region. On the eventual interfaces between two subdomains we impose discontinuous transition conditions (TCs). Finally, we have classical local BCs at the outer boundaries. Such problems are related to some heat transfer problems e.g. in a horizontal cross section of a wall enclosing an air cave.     

The Singular Sources Method for an Inverse Transmission Problem

We employ the singular sources method introduced in [4] to solve an inverse transmission scattering problem for the Helmholtz equation or D, respectively, where the total field u satisfies the transmission conditions on the boundary of some domain D with some constant β. The main idea of the singular sources scheme is to reconstruct the scattered field of point sources or higher multipoles (·, z) with source point z in its source point from the far field pattern of scattered plane waves. The function (z, z) is shown to become singular for z→∂D. This can be used to detect the shape D of the scattering object.Here, we will show how in addition to reconstructions of the shape D of the scattering object, the constant β can be reconstructed without solving the direct scattering problem. This extends the singular sources method from the reconstruction of geometric properties of an object to the reconstruction of physical quantities.     

Spine Based Shape Parameterisation for PDE Surfaces

The aim of this paper is to show how the spine of a PDE surface can be generated and how it can be used to efficiently parameterise a PDE surface. For the purpose of the work presented here an approximate analytic solution form for the chosen PDE is utilised. It is shown that the spine of the PDE surface is then computed as a by-product of this analytic solution. Furthermore, it is shown that a parameterisation can be introduced on the spine enabling intuitive manipulation of PDE surfaces.     

Image Synthesis for Inverse Obstacle Scattering Using the Eigenfunction Expansion Theorem

In recent years several new inverse scattering techniques have been developed that determine the boundary of an unknown obstacle by reconstructing the surrounding scattered field. In the case of sound soft obstacles, the boundary is usually found as the minimum contour of the total field. In this note we derive a different approach for imaging the boundary from the reconstructed fields based on a generalization of the eigenfunction expansion theorem. The aim of this alternative approach is the construction of higher contrast images than is currently obtained with the minimum contour approach.     

Surfaces parametrized by the normals

For a surface with non vanishing Gaussian curvature the Gauss map is regular and can be inverted. This makes it possible to use the normal as the parameter, and then it is trivial to calculate the normal and the Gauss map. This in turns makes it easy to calculate offsets, the principal curvatures, the principal directions, etc. Such a parametrization is not only a theoretical possibility but can be used concretely. One way of obtaining this parametrization is to specify the support function as a function of the normal, i.e., as a function on the unit sphere. The support function is the distance from the origin to the tangent plane and the surface is simply considered as the envelope of its family of tangent planes. Suppose we are given points and normals and we want a C ^k -surface interpolating these data. The data gives the value and gradients of the support function at certain points (the given normals) on the unit sphere, and the surface can be defined by determining the support function as a C ^k function interpolating the given values and gradients.     

A Semi-smooth Newton Method for Regularized State-constrained Optimal Control of the Navier-Stokes Equations

In this paper, we study semi-smooth Newton methods for the numerical solution of regularized pointwise state-constrained optimal control problems governed by the Navier-Stokes equations. After deriving an appropriate optimality system for the original problem, a class of Moreau-Yosida regularized problems is introduced and the convergence of their solutions to the original optimal one is proved. For each regularized problem a semi-smooth Newton method is applied and its local superlinear convergence verified. Finally, selected numerical results illustrate the behavior of the method and a comparison between the max-min and the Fischer-Burmeister as complementarity functionals is carried out.     

Tensor Product Multigrid for Maxwell's Equation with Aligned Anisotropy

When simulating electromagnetic phenomena in symmetric cavities, it is often possible to exploit the symmetry in order to reduce the dimension of the problem, thereby reducing the amount of work necessary for its numerical solution. Usually, this reduction leads not only to a much lower number of unknowns in the discretized system, but also changes the behaviour of the coefficients of the differential operator in an unfavourable way, usually leading to the transformed system being not elliptic with respect to norms corresponding to two-dimensional space, thus limiting the use of standard multigrid techniques. In this paper, we introduce a robust multigrid method for Maxwell's equation in two dimensions that is especially suited for coefficients resulting from coordinate transformations, i.e. that are aligned with the coordinate axes. Using a variant of the technique introduced in [5], we can prove robustness of the multigrid method for domains of tensor-product structure and coefficients depending on only one of the coordinates. Received July 17, 2000; revised October 27, 2000     

A Matrix-free Two-grid Preconditioner for Solving Boundary Integral Equations in Electromagnetism

In this paper, we describe a matrix-free iterative algorithm based on the GMRES method for solving electromagnetic scattering problems expressed in an integral formulation. Integral methods are an interesting alternative to differential equation solvers for this problem class since they do not require absorbing boundary conditions and they mesh only the surface of the radiating object giving rise to dense and smaller linear systems of equations. However, in realistic applications the discretized systems can be very large and for some integral formulations, like the popular Electric Field Integral Equation, they become ill-conditioned when the frequency increases. This means that iterative Krylov solvers have to be combined with fast methods for the matrix-vector products and robust preconditioning to be affordable in terms of CPU time. In this work we describe a matrix-free two-grid preconditioner for the GMRES solver combined with the Fast Multipole Method. The preconditioner is an algebraic two-grid cycle built on top of a sparse approximate inverse that is used as smoother, while the grid transfer operators are defined using spectral information of the preconditioned matrix. Experiments on a set of linear systems arising from real radar cross section calculation in industry illustrate the potential of the proposed approach for solving large-scale problems in electromagnetism.     

Solving the signorini problem on the basis of domain decomposition techniques

The finite element discretization of the Signorini Problem leads to a large scale constrained minimization problem. To improve the convergence rate of the projection method preconditioning must be developed. To be effective, the relative condition number of the system matrix with respect to the preconditioning matrix has to be small and the applications of the preconditioner as well as the projection onto the set of feasible elements have to be fast computable. In this paper, we show how to construct and analyze such preconditioners on the basis of domain decomposition techniques. The numerical results obtained for the Signorini problem as well as for contact problems in plane elasticity confirm the theoretical analysis quite well.     

A Hermitian Box-scheme for One-dimensional Elliptic Equations – Application to Problems with High Contrasts in the Ellipticity

We introduce a new box-scheme, called ``hermitian box-scheme' on the model of the one-dimensional Poisson problem. The scheme combines features of the box-scheme of Keller, [20], [13], with the hermitian approximation of the gradient on a compact stencil, which is characteristic of compact schemes, [9], [21]. The resulting scheme is proved to be 4th order accurate for the primitive unknown u and its gradient p. The proved convergence rate is 1.5 for (u,p) in the discrete L ² norm. The connection with a non standard mixed finite element method is given. Finally, numerical results are displayed on pertinent 1-D elliptic problems with high contrasts in the ellipticity, showing in practice convergence rates ranging from 1 to 2.5 in the discrete H ¹ norm. This work has been performed with the support of the GDR MOMAS, (ANDRA, CEA, EDF, BRGM and CNRS): Modélisation pour le stockage des déchets radioactifs. The author thanks especially A. Bourgeat for his encouragements and his interest in this work.     

Composite Finite Elements for Elliptic Boundary Value Problems with Discontinuous Coefficients

In this paper, we will introduce composite finite elements for solving elliptic boundary value problems with discontinuous coefficients. The focus is on problems where the geometry of the interfaces between the smooth regions of the coefficients is very complicated. On the other hand, efficient numerical methods such as, e.g., multigrid methods, wavelets, extrapolation, are based on a multi-scale discretization of the problem. In standard finite element methods, the grids have to resolve the structure of the discontinuous coefficients. Thus, straightforward coarse scale discretizations of problems with complicated coefficient jumps are not obvious. In this paper, we define composite finite elements for problems with discontinuous coefficients. These finite elements allow the coarsening of finite element spaces independently of the structure of the discontinuous coefficients. Thus, the multigrid method can be applied to solve the linear system on the fine scale. We focus on the construction of the composite finite elements and the efficient, hierarchical realization of the intergrid transfer operators. Finally, we present some numerical results for the multigrid method based on the composite finite elements (CFE–MG).     

Avoiding slave points in an adaptive refinement procedure for convection-diffusion problems in 2D

A finite difference method is presented for singularly perturbed convection-diffusion problems with discretization error estimate of nearly second order. In a standard patched adaptive refinement method certain slave nodes appear where the approximation is done by interpolating the values of the approximate solution at adjacent nodes. This deteriorates the accuracy of truncation error. In order to avoid the slave points we change the stencil at the interface points from a cross to a skew one. The efficiency of this technique is illustrated by numerical experiments in 2D.     

High-Order Time-Accurate Parallel Schemes for Parabolic Singularly Perturbed Problems with Convection

The first boundary value problem for a singularly perturbed parabolic equation of convection-diffusion type on an interval is studied. For the approximation of the boundary value problem we use earlier developed finite difference schemes, ɛ-uniformly of a high order of accuracy with respect to time, based on defect correction. New in this paper is the introduction of a partitioning of the domain for these ɛ-uniform schemes. We determine the conditions under which the difference schemes, applied independently on subdomains may accelerate (ɛ-uniformly) the solution of the boundary value problem without losing the accuracy of the original schemes. Hence, the simultaneous solution on subdomains can in principle be used for parallelization of the computational method. Received December 3, 1999; revised April 20, 2000