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Helmholtz Equation (helmholtz + equation)
Selected AbstractsMultifrequency Analysis for the Helmholtz EquationPROCEEDINGS IN APPLIED MATHEMATICS & MECHANICS, Issue 1 2003M. Köhl This paper is a brief review of a new numerical method for the multifrequency analysis of the three-dimensional Helmholtz equation. We describe the principles of this method which is based on the identity of the Fourier transform with respect to the wave number. Some numerical examples for the solution were presented at the oral session. [source] A study into the feasibility of using two parallel sparse direct solvers for the Helmholtz equation on Linux clustersCONCURRENCY AND COMPUTATION: PRACTICE & EXPERIENCE, Issue 7 2006G. Z. M. Berglund Abstract Two state-of-the-art parallel software packages for the direct solution of sparse linear systems based on LU-decomposition, MUMPS and SuperLU_DIST have been tested as black-box solvers on problems derived from finite difference discretizations of the Helmholtz equation. The target architecture has been Linux clusters, for which no consistent set of tests of the algorithms implemented in these packages has been published. The investigation consists of series of memory and time scalability checks and has focused on examining the applicability of the algorithms when processing very large sparse matrices on Linux cluster platforms. Special emphasis has been put on monitoring the behaviour of the packages when the equation systems need to be solved for multiple right-hand sides, which is the case, for instance, when modelling a seismic survey. The outcome of the tests points at poor efficiency of the tested algorithms during application of the LU-factors in the solution phase on this type of architecture, where the communication acts as an impasse. Copyright © 2005 John Wiley & Sons, Ltd. [source] Recovering acoustic reflectivity using Dirichlet-to-Neumann maps and left- and right-operating adjoint propagatorsGEOPHYSICAL JOURNAL INTERNATIONAL, Issue 1 2005M. W. P. Dillen SUMMARY Constructing an image of the Earth subsurface from acoustic wave reflections has previously been described as a recursive downward redatuming of sources and receivers. Most of the methods that have been presented involve reflectivity and propagators associated with one-way wavefield components. In this paper, we consider the reflectivity relation between two-way wavefield components, each a solution of a Helmholtz equation. To construct forward and inverse propagators, and a reflection operator, the invariant-embedding technique is followed, using Dirichlet-to-Neumann maps. Employing bilinear and sesquilinear forms, the forward- and inverse-scattering problems, respectively, are treated analogously. Through these mathematical constructs, the relationship between a causality radiation condition and symmetry, with respect to a bilinear form, is associated with the requirement of an anticausality radiation condition with respect to a sesquilinear form. Using reciprocity, sources and receivers are redatumed recursively to the reflector, employing left- and right-operating adjoint propagators. The exposition of the proposed method is formal, that is numerical applications are not derived. The key to applications lies in the explicit representation, characterization and approximation of the relevant operators (symbols) and fundamental solutions (path integrals). Existing constructive work which could be applied to the proposed method are referred to in the text. [source] P-wave and S-wave decomposition in boundary integral equation for plane elastodynamic problemsINTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN BIOMEDICAL ENGINEERING, Issue 12 2003Emmanuel Perrey-Debain Abstract The method of plane wave basis functions, a subset of the method of Partition of Unity, has previously been applied successfully to finite element and boundary element models for the Helmholtz equation. In this paper we describe the extension of the method to problems of scattering of elastic waves. This problem is more complicated for two reasons. First, the governing equation is now a vector equation and second multiple wave speeds are present, for any given frequency. The formulation has therefore a number of novel features. A full development of the necessary theory is given. Results are presented for some classical problems in the scattering of elastic waves. They demonstrate the same features as those previously obtained for the Helmholtz equation, namely that for a given level of error far fewer degrees of freedom are required in the system matrix. The use of the plane wave basis promises to yield a considerable increase in efficiency over conventional boundary element formulations in elastodynamics. Copyright © 2003 John Wiley & Sons, Ltd. [source] Coupling of mapped wave infinite elements and plane wave basis finite elements for the Helmholtz equation in exterior domainsINTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN BIOMEDICAL ENGINEERING, Issue 10 2003Rie Sugimoto Abstract The theory for coupling of mapped wave infinite elements and special wave finite elements for the solution of the Helmholtz equation in unbounded domains is presented. Mapped wave infinite elements can be applied to boundaries of arbitrary shape for exterior wave problems without truncation of the domain. Special wave finite elements allow an element to contain many wavelengths rather than having many finite elements per wavelength like conventional finite elements. Both types of elements include trigonometric functions to describe wave behaviour in their shape functions. However the wave directions between nodes on the finite element/infinite element interface can be incompatible. This is because the directions are normally globally constant within a special finite element but are usually radial from the ,pole' within a mapped wave infinite element. Therefore forcing the waves associated with nodes on the interface to be strictly radial is necessary to eliminate this internode incompatibility. The coupling of these elements was tested for a Hankel source problem and plane wave scattering by a cylinder and good accuracy was achieved. This paper deals with unconjugated infinite elements and is restricted to two-dimensional problems. Copyright © 2003 John Wiley & Sons, Ltd. [source] Dispersion analysis of the meshfree radial point interpolation method for the Helmholtz equationINTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN ENGINEERING, Issue 12 2009Christina Wenterodt Abstract When numerical methods such as the finite element method (FEM) are used to solve the Helmholtz equation, the solutions suffer from the so-called pollution effect which leads to inaccurate results, especially for high wave numbers. The main reason for this is that the wave number of the numerical solution disagrees with the wave number of the exact solution, which is known as dispersion. In order to obtain admissible results a very high element resolution is necessary and increased computational time and memory capacity are the consequences. In this paper a meshfree method, namely the radial point interpolation method (RPIM), is investigated with respect to the pollution effect in the 2D-case. It is shown that this methodology is able to reduce the dispersion significantly. Two modifications of the RPIM, namely one with polynomial reproduction and another one with a problem-dependent sine/cosine basis, are also described and tested. Numerical experiments are carried out to demonstrate the advantages of the method compared with the FEM. For identical discretizations, the RPIM yields considerably better results than the FEM. Copyright © 2008 John Wiley & Sons, Ltd. [source] Improved accuracy for the Helmholtz equation in unbounded domainsINTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN ENGINEERING, Issue 15 2004Eli Turkel Abstract Based on properties of the Helmholtz equation, we derive a new equation for an auxiliary variable. This reduces much of the oscillations of the solution leading to more accurate numerical approximations to the original unknown. Computations confirm the improved accuracy of the new models in both two and three dimensions. This also improves the accuracy when one wants the solution at neighbouring wavenumbers by using an expansion in k. We examine the accuracy for both waveguide and scattering problems as a function of k, h and the forcing mode l. The use of local absorbing boundary conditions is also examined as well as the location of the outer surface as functions of k. Connections with parabolic approximations are analysed. Copyright © 2004 John Wiley & Sons, Ltd. [source] Singularity extraction technique for integral equation methods with higher order basis functions on plane triangles and tetrahedraINTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN ENGINEERING, Issue 8 2003Seppo Järvenpää Abstract A numerical solution of integral equations typically requires calculation of integrals with singular kernels. The integration of singular terms can be considered either by purely numerical techniques, e.g. Duffy's method, polar co-ordinate transformation, or by singularity extraction. In the latter method the extracted singular integral is calculated in closed form and the remaining integral is calculated numerically. This method has been well established for linear and constant shape functions. In this paper we extend the method for polynomial shape functions of arbitrary order. We present recursive formulas by which we can extract any number of terms from the singular kernel defined by the fundamental solution of the Helmholtz equation, or its gradient, and integrate the extracted terms times a polynomial shape function in closed form over plane triangles or tetrahedra. The presented formulas generalize the singularity extraction technique for surface and volume integral equation methods with high-order basis functions. Numerical experiments show that the developed method leads to a more accurate and robust integration scheme, and in many cases also a faster method than, for example, Duffy's transformation. Copyright © 2003 John Wiley & Sons, Ltd. [source] A fictitious domain decomposition method for the solution of partially axisymmetric acoustic scattering problems.INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN ENGINEERING, Issue 1 2003Part 2: Neumann boundary conditions Abstract We present a fictitious domain decomposition method for the fast solution of acoustic scattering problems characterized by a partially axisymmetric sound-hard scatterer. We apply this method to the solution of a mock-up submarine problem, and highlight its computational advantages and intrinsic parallelism. A key component of our method is an original idea for addressing a Neumann boundary condition in the general framework of a fictitious domain method. This idea is applicable to many other linear partial differential equations besides the Helmholtz equation. Copyright © 2003 John Wiley & Sons, Ltd. [source] An iterative defect-correction type meshless method for acousticsINTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN ENGINEERING, Issue 15 2003V. Lacroix Abstract Accurate numerical simulation of acoustic wave propagation is still an open problem, particularly for medium frequencies. We have thus formulated a new numerical method better suited to the acoustical problem: the element-free Galerkin method (EFGM) improved by appropriate basis functions computed by a defect correction approach. One of the EFGM advantages is that the shape functions are customizable. Indeed, we can construct the basis of the approximation with terms that are suited to the problem which has to be solved. Acoustical problems, in cavities , with boundary T, are governed by the Helmholtz equation completed with appropriate boundary conditions. As the pressure p(x,y) is a complex variable, it can always be expressed as a function of cos,(x,y) and sin,(x,y) where ,(x,y) is the phase of the wave in each point (x,y). If the exact distribution ,(x,y) of the phase is known and if a meshless basis {1, cos,(x,y), sin, (x,y) } is used, then the exact solution of the acoustic problem can be obtained. Obviously, in real-life cases, the distribution of the phase is unknown. The aim of our work is to resolve, as a first step, the acoustic problem by using a polynomial basis to obtain a first approximation of the pressure field p(x,y). As a second step, from p(x,y) we compute the distribution of the phase ,(x,y) and we introduce it in the meshless basis in order to compute a second approximated pressure field p(x,y). From p(x,y), a new distribution of the phase is computed in order to obtain a third approximated pressure field and so on until a convergence criterion, concerning the pressure or the phase, is obtained. So, an iterative defect-correction type meshless method has been developed to compute the pressure field in ,. This work will show the efficiency of this meshless method in terms of accuracy and in terms of computational time. We will also compare the performance of this method with the classical finite element method. Copyright © 2003 John Wiley & Sons, Ltd. [source] Fast direct solution of the Helmholtz equation with a perfectly matched layer or an absorbing boundary conditionINTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN ENGINEERING, Issue 14 2003Erkki Heikkola Abstract We consider the efficient numerical solution of the Helmholtz equation in a rectangular domain with a perfectly matched layer (PML) or an absorbing boundary condition (ABC). Standard bilinear (trilinear) finite-element discretization on an orthogonal mesh leads to a separable system of linear equations for which we describe a cyclic reduction-type fast direct solver. We present numerical studies to estimate the reflection of waves caused by an absorbing boundary and a PML, and we optimize certain parameters of the layer to minimize the reflection. Copyright © 2003 John Wiley & Sons, Ltd. [source] A numerical integration scheme for special finite elements for the Helmholtz equationINTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN ENGINEERING, Issue 4 2003Peter Bettess Abstract The theory for integrating the element matrices for rectangular, triangular and quadrilateral finite elements for the solution of the Helmholtz equation for very short waves is presented. A numerical integration scheme is developed. Samples of Maple and Fortran code for the evaluation of integration abscissæ and weights are made available. The results are compared with those obtained using large numbers of Gauss,Legendre integration points for a range of testing wave problems. The results demonstrate that the method gives correct results, which gives confidence in the procedures, and show that large savings in computation time can be achieved. Copyright © 2002 John Wiley & Sons, Ltd. [source] The performance of spheroidal infinite elementsINTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN ENGINEERING, Issue 12 2001R. J. Astley Abstract A number of spheroidal and ellipsoidal infinite elements have been proposed for the solution of unbounded wave problems in the frequency domain, i.e solutions of the Helmholtz equation. These elements are widely believed to be more effective than conventional spherical infinite elements in cases where the radiating or scattering object is slender or flat and can therefore be closely enclosed by a spheroidal or an ellipsoidal surface. The validity of this statement is investigated in the current article. The radial order which is required for an accurate solution is shown to depend strongly not only upon the type of element that is used, but also on the aspect ratio of the bounding spheroid and the non-dimensional wave number. The nature of this dependence can partially be explained by comparing the non-oscillatory component of simple source solutions to the terms available in the trial solution of spheroidal elements. Numerical studies are also presented to demonstrate the rates at which convergence can be achieved, in practice, by unconjugated-(,Burnett') and conjugated (,Astley-Leis')-type elements. It will be shown that neither formulation is entirely satisfactory at high frequencies and high aspect ratios. Copyright © 2001 John Wiley & Sons, Ltd. [source] A cascadic conjugate gradient algorithm for mass conservative, semi-implicit discretization of the shallow water equations on locally refined structured gridsINTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN FLUIDS, Issue 1-2 2002Luca Bonaventura Abstract A semi-implicit, mass conservative discretization scheme is applied to the two-dimensional shallow water equations on a hierarchy of structured, locally refined Cartesian grids. Different resolution grids are fully interacting and the discrete Helmholtz equation obtained from the semi-implicit discretization is solved by the cascadic conjugate gradient method. A flux correction is applied at the interface between the coarser and finer discretization grids, so as to ensure discrete mass conservation, along with symmetry and diagonal dominance of the resulting matrix. Two-dimensional idealized simulations are presented, showing the accuracy and the efficiency of the resulting method. Copyright © 2002 John Wiley & Sons, Ltd. [source] Multi-periodic eigensolutions to the Dirac operator and applications to the generalized Helmholtz equation on flat cylinders and on the n -torusMATHEMATICAL METHODS IN THE APPLIED SCIENCES, Issue 16 2009Denis Constales Abstract In this paper, we study the solutions to the generalized Helmholtz equation with complex parameter on some conformally flat cylinders and on the n -torus. Using the Clifford algebra calculus, the solutions can be expressed as multi-periodic eigensolutions to the Dirac operator associated with a complex parameter ,,,. Physically, these can be interpreted as the solutions to the time-harmonic Maxwell equations on these manifolds. We study their fundamental properties and give an explicit representation theorem of all these solutions and develop some integral representation formulas. In particular, we set up Green-type formulas for the cylindrical and toroidal Helmholtz operator. As a concrete application, we explicitly solve the Dirichlet problem for the cylindrical Helmholtz operator on the half cylinder. Finally, we introduce hypercomplex integral operators on these manifolds, which allow us to represent the solutions to the inhomogeneous Helmholtz equation with given boundary data on cylinders and on the n -torus. Copyright © 2009 John Wiley & Sons, Ltd. [source] Scattering from infinite rough tubular surfacesMATHEMATICAL METHODS IN THE APPLIED SCIENCES, Issue 4 2007Xavier Claeys Abstract We study the Helmholtz equation in the exterior of an infinite perturbed cylinder with a Dirichlet boundary condition. Existence and uniqueness of solutions are established using the variational technique introduced (SIAM J. Math. Anal. 2005; 37(2):598,618). We also provide stability estimates with explicit dependence of the constants in terms of the frequency and the perturbed cylinder thickness. Copyright © 2006 John Wiley & Sons, Ltd. [source] Integral equation methods for scattering by infinite rough surfacesMATHEMATICAL METHODS IN THE APPLIED SCIENCES, Issue 6 2003Bo Zhang Abstract In this paper, we consider the Dirichlet and impedance boundary value problems for the Helmholtz equation in a non-locally perturbed half-plane. These boundary value problems arise in a study of time-harmonic acoustic scattering of an incident field by a sound-soft, infinite rough surface where the total field vanishes (the Dirichlet problem) or by an infinite, impedance rough surface where the total field satisfies a homogeneous impedance condition (the impedance problem). We propose a new boundary integral equation formulation for the Dirichlet problem, utilizing a combined double- and single-layer potential and a Dirichlet half-plane Green's function. For the impedance problem we propose two boundary integral equation formulations, both using a half-plane impedance Green's function, the first derived from Green's representation theorem, and the second arising from seeking the solution as a single-layer potential. We show that all the integral equations proposed are uniquely solvable in the space of bounded and continuous functions for all wavenumbers. As an important corollary we prove that, for a variety of incident fields including an incident plane wave, the impedance boundary value problem for the scattered field has a unique solution under certain constraints on the boundary impedance. Copyright © 2003 John Wiley & Sons, Ltd. [source] On the diffraction of Poincaré wavesMATHEMATICAL METHODS IN THE APPLIED SCIENCES, Issue 12 2001P. A. Martin Abstract The diffraction of tidal waves (Poincaré waves) by islands and barriers on water of constant finite depth is governed by the two-dimensional Helmholtz equation. One effect of the Earth's rotation is to complicate the boundary condition on rigid boundaries: a linear combination of the normal and tangential derivatives is prescribed. (This would be an oblique derivative if the coefficients were real.) Corresponding boundary-value problems are treated here using layer potentials, generalizing the usual approach for the standard exterior boundary-value problems of acoustics. Singular integral equations are obtained for islands (scatterers with non-empty interiors) whereas hypersingular integral equations are obtained for thin barriers. Copyright © 2001 John Wiley & Sons, Ltd. [source] On the stability and convergence of the finite section method for integral equation formulations of rough surface scatteringMATHEMATICAL METHODS IN THE APPLIED SCIENCES, Issue 4 2001A. Meier We consider the Dirichlet and Robin boundary value problems for the Helmholtz equation in a non-locally perturbed half-plane, modelling time harmonic acoustic scattering of an incident field by, respectively, sound-soft and impedance infinite rough surfaces. Recently proposed novel boundary integral equation formulations of these problems are discussed. It is usual in practical computations to truncate the infinite rough surface, solving a boundary integral equation on a finite section of the boundary, of length 2A, say. In the case of surfaces of small amplitude and slope we prove the stability and convergence as A,, of this approximation procedure. For surfaces of arbitrarily large amplitude and/or surface slope we prove stability and convergence of a modified finite section procedure in which the truncated boundary is ,flattened' in finite neighbourhoods of its two endpoints. Copyright © 2001 John Wiley & Sons, Ltd. [source] Nonconforming Galerkin methods for the Helmholtz equationNUMERICAL METHODS FOR PARTIAL DIFFERENTIAL EQUATIONS, Issue 5 2001Jim Douglas Jr. Abstract Nonconforming Galerkin methods for a Helmholtz-like problem arising in seismology are discussed both for standard simplicial linear elements and for several new rectangular elements related to bilinear or trilinear elements. Optimal order error estimates in a broken energy norm are derived for all elements and in L2 for some of the elements when proper quadrature rules are applied to the absorbing boundary condition. Domain decomposition iterative procedures are introduced for the nonconforming methods, and their convergence at a predictable rate is established. © 2001 John Wiley & Sons, Inc. Numer Methods Partial Differential Eq 17: 475,494, 2001 [source] Multifrequency Analysis for the Helmholtz EquationPROCEEDINGS IN APPLIED MATHEMATICS & MECHANICS, Issue 1 2003M. Köhl This paper is a brief review of a new numerical method for the multifrequency analysis of the three-dimensional Helmholtz equation. We describe the principles of this method which is based on the identity of the Fourier transform with respect to the wave number. Some numerical examples for the solution were presented at the oral session. [source] Cloaking via change of variables for the Helmholtz equation in the whole spaceCOMMUNICATIONS ON PURE & APPLIED MATHEMATICS, Issue 11 2010Hoai-Minh Nguyen This paper is devoted to the study of a cloaking device that is composed of a standard near cloak based on a regularization of the transformation optics, i.e., a change of variables that blows up a small ball to the cloaked region, and a fixed lossy layer for the Helmholtz equation in the whole space of dimension 2 or 3 with the outgoing condition at infinity. We establish a degree of near invisibility, which is independent of the content inside the cloaked region, for this device. We also show that the lossy layer is necessary to ensure the validity of the degree of near invisibility when no constraint on physical properties inside the cloaked region is imposed. © 2010 Wiley Periodicals, Inc. [source] Moisture sorption isotherms and thermodynamic properties of apple Fuji and garlicINTERNATIONAL JOURNAL OF FOOD SCIENCE & TECHNOLOGY, Issue 10 2008Mariana A. Moraes Summary The moisture equilibrium isotherms of garlic and apple were determined at 50, 60 and 70 °C using the gravimetric static method. The experimental data were analysed using GAB, BET, Henderson,Thompson and Oswin equations. The isosteric heat and the differential entropy of desorption were determined by applying Clausius,Clapeyron and Gibbs,Helmholtz equations, respectively. The GAB equation showed the best fitting to the experimental data (R2 > 99% and E% < 10%). The monolayer moisture content values for apple were higher than those for garlic at the studied temperatures; the values varied from 0.050 to 0.056 and from 0.107 to 0.168 for garlic and apple, respectively. The isosteric heat and the differential entropy of desorption were estimated in function of the moisture content. The values of these thermodynamic properties were higher for apple (in range 48,100 kJ mol,1 and 14,150 J mol,1 K,1) than for garlic (in range 43,68 kJ mol,1 and 0,66 J mol,1 K,1). The water surface area values decreased with increasing temperature. The Kelvin and the Halsey equations were used to calculate the pore size distribution. [source] Convergence analysis of a balancing domain decomposition method for solving a class of indefinite linear systemsNUMERICAL LINEAR ALGEBRA WITH APPLICATIONS, Issue 9 2009Jing Li Abstract A variant of balancing domain decomposition method by constraints (BDDC) is proposed for solving a class of indefinite systems of linear equations of the form (K,,2M)u=f, which arise from solving eigenvalue problems when an inverse shifted method is used and also from the finite element discretization of Helmholtz equations. Here, both K and M are symmetric positive definite. The proposed BDDC method is closely related to the previous dual,primal finite element tearing and interconnecting method (FETI-DP) for solving this type of problems (Appl. Numer. Math. 2005; 54:150,166), where a coarse level problem containing certain free-space solutions of the inherent homogeneous partial differential equation is used in the algorithm to accelerate the convergence. Under the condition that the diameters of the subdomains are small enough, the convergence rate of the proposed algorithm is established, which depends polylogarithmically on the dimension of the individual subdomain problems and which improves with a decrease of the subdomain diameters. These results are supported by numerical experiments of solving a two-dimensional problem. Copyright © 2009 John Wiley & Sons, Ltd. [source] GPU-accelerated boundary element method for Helmholtz' equation in three dimensionsINTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN ENGINEERING, Issue 10 2009Toru Takahashi Abstract Recently, the application of graphics processing units (GPUs) to scientific computations is attracting a great deal of attention, because GPUs are getting faster and more programmable. In particular, NVIDIA's GPUs called compute unified device architecture enable highly mutlithreaded parallel computing for non-graphic applications. This paper proposes a novel way to accelerate the boundary element method (BEM) for three-dimensional Helmholtz' equation using CUDA. Adopting the techniques for the data caching and the double,single precision floating-point arithmetic, we implemented a GPU-accelerated BEM program for GeForce 8-series GPUs. The program performed 6,23 times faster than a normal BEM program, which was optimized for an Intel's quad-core CPU, for a series of boundary value problems with 8000,128000 unknowns, and it sustained a performance of 167,Gflop/s for the largest problem (1 058 000 unknowns). The accuracy of our BEM program was almost the same as that of the regular BEM program using the double precision floating-point arithmetic. In addition, our BEM was applicable to solve realistic problems. In conclusion, the present GPU-accelerated BEM works rapidly and precisely for solving large-scale boundary value problems for Helmholtz' equation. Copyright © 2009 John Wiley & Sons, Ltd. [source] An integral formulation procedure for the solutions to Helmholtz's equation in spherically symmetric mediaMATHEMATICAL METHODS IN THE APPLIED SCIENCES, Issue 11 2010Giacomo Caviglia Abstract Starting from Helmholtz's equation in inhomogeneous media, the associated radial second-order equation is investigated through a Volterra integral equation. First the integral equation is considered in a sphere. Boundedness, uniqueness and existence of the (regular) solution are established and the series form of the solution is provided. An estimate is determined for the error arising when the series is truncated. Next the analogous problem is considered for a spherical layer. Again, boundedness, uniqueness and existence of two base solutions are established and error estimates are determined. The procedure proves more effective in the sphere. Copyright © 2009 John Wiley & Sons, Ltd. [source] | ||||||||||||||||