Numerical solution of electromagnetic scattering from a large partly covered cavity

The paper focuses on the numerical study of electromagnetic scattering from two-dimensional (2D) large partly covered cavities, which is described by the Helmholtz equation with a nonlocal boundary condition on the aperture. The classical five-point finite difference method is applied for the discretization of the Helmholtz equation and a linear approximation is used for the nonlocal boundary condition. We prove the existence and uniqueness of the numerical solution when the medium in the cavity is y-direction layered or the number of the mesh points on the aperture is large enough. The fast algorithm proposed in Bao and Sun (2005) [2] for open cavity models is extended to solving the partly covered cavity problem with (vertically) layered media. A preconditioned Krylov subspace method is proposed to solve the partly covered cavity problem with a general medium, in which a layered medium model is used as a preconditioner of the general model. Numerical results for several types of partly covered cavities with different wave numbers are reported and compared with those by ILU-type preconditioning algorithms. Our numerical experiments show that the proposed preconditioning algorithm is more efficient for partly covered cavity problems, particularly with large wave numbers.     

A Second-Order Method for the Electromagnetic Scattering from a Large Cavity

In this paper,we study the electromagnetic scattering from a two dimen- sional large rectangular open cavity embedded in an infinite ground plane,which is modelled by Helmholtz equations.By introducing nonlocal transparent boundary con- ditions,the problem in the open cavity is reduced to a bounded domain problem.A hypersingular integral operator and a weakly singular integral operator are involved in the TM and TE cases,respectively.A new second-order Toeplitz type approximation and a second-order finite difference scheme are proposed for approximating the hyper- singular integral operator on the aperture and the Helmholtz in the cavity,respectively. The existence and uniqueness of the numerical solution in the TE case are established for arbitrary wavenumbers.A fast algorithm for the second-order approximation is pro- posed for solving the cavity model with layered media.Numerical results show the second-order accuracy and efficiency of the fast algorithm.More important is that the algorithm is easy to implement as a preconditioner for cavity models with more general media.     

A Moving Point Method for Arbitrary Peclet Number Multi-Dimensional Convection--Diffusion Equations

The moving point algorithm of Garder, Peaceman & Pozzi (SPEJ,March 1964, 26–36) will solve high Peclet number, multi-dimensional,convection-diffusion equations without numerical diffusion orspurious oscillations. The method uses a moving set of pointstogether with a fixed Eulerian mesh and avoids the mesh tanglingproblems of many moving mesh methods. However, numerical resultsof Price, Cavendish & Varga (SPEJ, Sept. 1968, 293–303)indicate that with a fixed number of moving points per meshcell the algorithm does not converge under mesh refinement.These authors also show, for fixed mesh lengths and a fixednumber of moving points per cell, that the moving point methodis more accurate than conventional fixed-mesh algorithms ifthe Peclet number is sufficiently large. We introduce and analysis a moving point algorithm which weprove to converge for arbitrary Peclet numbers. Numerical resultsdemonstrating the convergence and the accuracy of the movingpoint method relative to standard finite difference algorithmsare described.     

High Performance Computing Techniques for the FEM Simulation in Structural Mechanics

For the solution of large scale simulations in structural mechanics iterative solving methods are mandatory. The efficiency of such methods can crucially depend on different factors: choice of material parameters, quality of the underlying computational mesh and number of processors in a parallel computing system. We distinguish between three aspects of ‘efficiency’: processor efficiency (degree to which the solving algorithm is able to exploit the processor's computational power), parallel efficiency (ratio between computation and communication times) and numerical efficiency (convergence behaviour). With the new FEM software package Feast we pursue the aim to develop a solver mechanism which at the same time gains high efficiencies in all three aspects, while trying to minimise the mentioned dependencies. (© 2006 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Numerical approximation of the Smagorinsky model by a two-level method

A two-level method for discretizing the Smagorinsky model for the numerical simulation of turbulent flows is proposed. In the two-level algorithm, the solution to the fully nonlinear coarse mesh problem is utilized in a single-step linear fine mesh problem. When modeling parameters are chosen appropriately, the error in the two-level algorithm is comparable to the error in solving the fully nonlinear problem on the fine mesh. We provide an a priori error estimate for the two-level method, which yields appropriate scalings between the coarse and fine mesh-sizes (H and h, respectively), and the radius of the spatial filter used in the Smagorinsky model (δ). In addition, we provide an algorithm in which a coarse mesh correction is performed to further enhance the accuracy. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

One-dimensional center-based l 1-clustering method

Motivated by the method for solving center-based Least Squares—clustering problem (Kogan in Introduction to clustering large and high-dimensional data, Cambridge University Press, 2007; Teboulle in J Mach Learn Res 8:65–102, 2007) we construct a very efficient iterative process for solving a one-dimensional center-based l ₁—clustering problem, on the basis of which it is possible to determine the optimal partition. We analyze the basic properties and convergence of our iterative process, which converges to a stationary point of the corresponding objective function for each choice of the initial approximation. Given is also a corresponding algorithm, which in only few steps gives a stationary point and the corresponding partition. The method is illustrated and visualized on the example of looking for an optimal partition with two clusters, where we check all stationary points of the corresponding minimizing functional. Also, the method is tested on the basis of large numbers of data points and clusters and compared with the method for solving the center-based Least Squares—clustering problem described in Kogan (2007) and Teboulle (2007).     

On Two Iterative Least-Squares Finite Element Schemes for the Incompressible Navier–Stokes Problem

This paper is mainly devoted to a comparative study of two iterative least-squares finite element schemes for solving the stationary incompressible Navier–Stokes equations with velocity boundary condition. Introducing vorticity as an additional unknown variable, we recast the Navier–Stokes problem into a first-order quasilinear velocity–vorticity–pressure system. Two Picard-type iterative least-squares finite element schemes are proposed to approximate the solution to the nonlinear first-order problem. In each iteration, we adopt the usual L ² least-squares scheme or a weighted L ² least-squares scheme to solve the corresponding Oseen problem and provide error estimates. We concentrate on two-dimensional model problems using continuous piecewise polynomial finite elements on uniform meshes for both iterative least-squares schemes. Numerical evidences show that the iterative L ² least-squares scheme is somewhat suitable for low Reynolds number flow problems, whereas for flows with relatively higher Reynolds numbers the iterative weighted L ² least-squares scheme seems to be better than the iterative L ² least-squares scheme. Numerical simulations of the two-dimensional driven cavity flow are presented to demonstrate the effectiveness of the iterative least-squares finite element approach.     

Two‐level Newton iterative method for the 2D/3D steady Navier‐Stokes equations

A combination method of the Newton iteration and two‐level finite element algorithm is applied for solving numerically the steady Navier‐Stokes equations under the strong uniqueness condition. This algorithm is motivated by applying the m Newton iterations for solving the Navier‐Stokes problem on a coarse grid and computing the Stokes problem on a fine grid. Then, the uniform stability and convergence with respect to ν of the two‐level Newton iterative solution are analyzed for the large m and small H and h << H. Finally, some numerical tests are made to demonstrate the effectiveness of the method. © 2011 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2012     

Application of a Krylov subspace iterative method in a multi-level adaptive technique to solve the mild-slope equation in nearshore regions

A multi-level adaptive numerical technique is applied to a nonlinear formulation of the mild-slope equation, to obtain the nearshore wave field, where the dominant processes of wave transformation are shoaling, refraction and diffraction. The advantage of this formulation over the traditional elliptic, parabolic and hyperbolic formulations is to require a lower minimum number of grid nodes per wavelength, thus, its capacity to predict the wave field for larger coastal areas. The efficiency of the interactions between the grid mesh levels, where two robust Krylov subspace iterative methods, the Bi-CGSTAB and the GMRES, are applied to solve the governing equation, is tested, for several hierarchies of grid mesh levels. The results show that the multi-level adaptive technique is efficient only if the GMRES iterative method is applied, and that for six grid mesh levels good results can be achieved for a residual as low as 10⁻³ for the finest grid.     

Anisotropic Wilson element with conforming finite element approximation for a coupled continuum pipe‐flow/Darcy model in Karst aquifers

Numerical method for a coupled continuum pipe‐flow/Darcy model describing flow in porous media with an embedded conduit pipe is considered. Wilson element on anisotropic mesh is used to solve the Darcy equation on porous matrix. The existence and uniqueness of the approximation solution are obtained. Optimal error estimates in L² and H¹ norms are established independent of the regularity condition on the mesh. Numerical examples show the efficiency of the presented scheme. With the same number of nodal points, the results using Wilson element on anisotropic mesh are much better than those of the same element and Q₁ element on regular mesh. Copyright © 2014 John Wiley & Sons, Ltd.     

On the scattering of spherical electromagnetic waves by a layered sphere

A spherical electromagnetic wave is scattered by a layered sphere.The exact expressions of the scattered and interior fields areobtained by solving the corresponding boundary-value problem,by means of a combination of Sommerfeld's and T-matrix methods.A recursive algorithm with respect to the number of layers isextracted for the computation of the fields in every layer.The far-field pattern and the scattering cross-sections aredetermined in terms of the physical and geometrical characteristicsof the scatterer. As the point-source tends to infinity, theknown results for plane wave incidence are recovered. Numericalresults are presented for several cases and various parametersof the layered spherical scatterer.     

A quadratic approximation method for minimizing a class of quasidifferentiable functions

Summary An unconstrained nonlinear programming problem with nondifferentiabilities is considered. The nondifferentiabilities arise from terms of the form max [f ₁(x), ...,f _n (x)], which may enter nonlinearly in the objective function. Local convex polyhedral upper approximations to the objective function are introduced. These approximations are used in an iterative method for solving the problem. The algorithm proceeds by solving quadratic programming subproblems to generate search directions. Approximate line searches ensure global convergence of the method to stationary points. The algorithm is conceptually simple and easy to implement. It generalizes efficient variable metric methods for minimax calculations.     

A new numerical method for the solution of the Boltzmann equation in the semiconductor nonlinear electron transport problem

A new iterative method for solving the Boltzmann transport equation in the space-uniform case is introduced. The method is based on the use of a mesh in the momentum space to represent the distribution function. In intermediate points the function is found with the help of interpolation. The method is used to study the hot-electron transport in bulk n-InN, which is a promising material for optoelectronic applications.     

Domain decomposition iterative procedures for solving scalar waves in the frequency domain

The propagation of dispersive waves can be modeled relevantly in the frequency domain. A wave problem in the frequency domain is difficult to solve numerically. In addition to having a complex–valued solution, the problem is neither Hermitian symmetric nor coercive in a wide range of applications in Geophysics or Quantum–Mechanics. In this paper, we consider a parallel domain decomposition iterative procedure for solving the problem by finite differences or conforming finite element methods. The analysis includes the decomposition of the domain into either the individual elements or larger subdomains ( of finite elements). To accelerate the speed of convergence, we introduce relaxation parameters on the subdomain interfaces and an artificial damping iteration. The convergence rate of the resulting algorithm turns out to be independent on the mesh size and the wave number. Numerical results carried out on an nCUBE2 parallel computer are presented to show the effectiveness of the method. Received October 30, 1995 / Revised version received January 10, 1997     

A weak Galerkin mixed finite element method for the Helmholtz equation with large wave numbers

In this article, a new weak Galerkin mixed finite element method is introduced and analyzed for the Helmholtz equation with large wave numbers. The stability and well‐posedness of the method are established for any wave number k without mesh size constraint. Allowing the use of discontinuous approximating functions makes weak Galerkin mixed method highly flexible in term of little restrictions on approximations and meshes. In the weak Galerkin mixed finite element formulation, approximation functions can be piecewise polynomials with different degrees on different elements and meshes can consist elements with different shapes. Suboptimal order error estimates in both discrete H¹ and L² norms are established for the weak Galerkin mixed finite element solutions. Numerical examples are tested to support the theory.     

A numerical method for solving an inverse thermoacoustic problem

In this paper, we consider an inverse problem of determining the initial condition of an initial boundary value problem for the wave equation with some additional information about solving a direct initial boundary value problem. The information is obtained from measurements at the boundary of the solution domain. The purpose of our paper is to construct a numerical algorithm for solving the inverse problem by an iterative method called a method of simple iteration (MSI) and to study the resolution quality of the inverse problem as a function of the number and location of measurement points. Three two-dimensional inverse problem formulations are considered. The results of our numerical calculations are presented. It is shown that the MSI decreases the objective functional at each iteration step. However, due to the ill-posedness of the inverse problem the difference between the exact and approximate solutions decreases up to some fixed number k _min, and then monotonically increases. This shows the regularizing properties of the MSI, and the iteration number can be considered a regularization parameter.     

A greedy algorithm for partition of unity collocation method in pricing American options

A greedy algorithm in combination with radial basis functions partition of unity collocation (GRBF‐PUC) scheme is used as a locally meshless method for American option pricing. The radial basis function partition of unity method (RBF‐PUM) is a localization technique. Because of having interpolation matrices with large condition numbers, global approximants and some local ones suffer from instability. To overcome this, a greedy algorithm is added to RBF‐PUM. The greedy algorithm furnishes a subset of best nodes among the points X. Such nodes are then used as points of trial in a locally supported RBF approximant for each partition. Using of greedy selected points leads to decreasing the condition number of interpolation matrices and reducing the burdensome in pricing American options.     

Parallel Solution of the Helmholtz Equation in a Multilayer Domain

We study time-harmonic wave propagation in layered, heterogeneous media. Solving this relatively complex application problem numerically is a challenging task. The full potential of algorithms, parallel programming models and computer architectures must be exploited. Our aim is to give a broad perspective on the various considerations that come into play. The basic parts of our algorithms consist of finite difference discretizations, domain decomposition and preconditioned iterative methods. We present two serial algorithms with different properties. Then, we discuss parallelization strategies using a local memory model, a shared memory model, or a combination of the two. The numerical experiments highlight the differences between the approaches and show results for three different combinations of algorithm and computer architecture that lead to viable solution methods.     

An algorithm for coarsening unstructured meshes

Summary. We develop and analyze a procedure for creating a hierarchical basis of continuous piecewise linear polynomials on an arbitrary, unstructured, nonuniform triangular mesh. Using these hierarchical basis functions, we are able to define and analyze corresponding iterative methods for solving the linear systems arising from finite element discretizations of elliptic partial differential equations. We show that such iterative methods perform as well as those developed for the usual case of structured, locally refined meshes. In particular, we show that the generalized condition numbers for such iterative methods are of order , where is the number of hierarchical basis levels. Received December 5, 1994     

Reconstructing initial data using observers: error analysis of the semi-discrete and fully discrete approximations

A new iterative algorithm for solving initial data inverse problems from partial observations has been recently proposed in Ramdani et al. (Automatica 46(10), 1616–1625, 2010). Based on the concept of observers (also called Luenberger observers), this algorithm covers a large class of abstract evolution PDE’s. In this paper, we are concerned with the convergence analysis of this algorithm. More precisely, we provide a complete numerical analysis for semi-discrete (in space) and fully discrete approximations derived using finite elements in space and an implicit Euler method in time. The analysis is carried out for abstract Schr?dinger and wave conservative systems with bounded observation (locally distributed).