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Quadratic spline collocation methods are formulated for the numerical solution of the Helmholtz equation in the unit square subject to non-homogeneous Dirichlet, Neumann and mixed boundary conditions, and also periodic boundary conditions. The methods are constructed so that they are: (a) of optimal accuracy, and (b) compact that is, the collocation equations can be solved using a matrix decomposition algorithm involving only tridiagonal linear systems. Using fast Fourier transforms, the computational cost of such an algorithm is O(N2logN) on an N×N uniform partition of the unit square. The results of numerical experiments demonstrate the optimal global accuracy of the methods as well as superconvergence phenomena. In particular, it is shown that the methods are fourth-order accurate at the nodes of the partition. © 2011 Elsevier Inc. 230 8 2880 2895 Cited By :8

We provide an overview of matrix decomposition algorithms (MDAs) for the solution of systems of linear equations arising when various discretization techniques are applied in the numerical solution of certain separable elliptic boundary value problems in the unit square. An MDA is a direct method which reduces the algebraic problem to one of solving a set of independent one-dimensional problems which are generally banded, block tridiagonal, or almost block diagonal. Often, fast Fourier transforms (FFTs) can be employed in an MDA with a resulting computational cost of O(N 2 logN) on an N?×?N uniform partition of the unit square. To formulate MDAs, we require knowledge of the eigenvalues and eigenvectors of matrices arising in corresponding two---point boundary value problems in one space dimension. In many important cases, these eigensystems are known explicitly, while in others, they must be computed. The first MDAs were formulated almost fifty years ago, for finite difference methods. Herein, we discuss more recent developments in the formulation and application of MDAs in spline collocation, finite element Galerkin and spectral methods, and the method of fundamental solutions. For ease of exposition, we focus primarily on the Dirichlet problem for Poisson's equation in the unit square, sketch extensions to other boundary conditions and to more involved elliptic problems, including the biharmonic Dirichlet problem, and report extensions to three dimensional problems in a cube. MDAs have also been used extensively as preconditioners in iterative methods for solving linear systems arising from discretizations of non-separable boundary value problems.

We propose a new geometric modelling method based on the so-called potential field (PF) modelling technique. The harmonic problem associated with this technique is solved numerically using the method of fundamental solutions (MFS). We investigate the applicability of the proposed approach to parametrically defined curves of varying complexity. Based on the MFS, we also provide definitions of the Boolean operations associated with the geometric modelling. Finally, we give practical applications of the method to computer-aided design and manufacturing problems. Copyright 2006 John Wiley & Sons, Ltd. 68 12 1257 1280 Cited By :18

The method of fundamental solutions (MFS) is applied to acoustic scattering and radiation for axisymmetric bodies and boundary conditions. The fundamental solution of the governing equation and its normal derivative, which are required in the formulation of the MFS, can be expressed in terms of complete elliptic integrals, which are evaluated using library software. The method is tested on several problems from the literature and the results compared with existing solutions. Numerical experiments demonstrate that the fictitious eigenfrequency problem which is encountered with the boundary element method is not present in the MFS. The method of fundamental solutions (MFS) was used to address axisymmetric acoustic scattering and radiation problems. The fundamental solutions of the governing equations involved and their normal derivatives required in the formulation of the MFS were calculated using existing library software and numerical integration. Application of the method to a number of problems from the literature yielded very good results. Further, it was found that the fictitious eigenfrequency difficulties encountered when using the boundary element method (BEM) are not present in the MFS. 104 6 3212 3218 Cited By :17

We formulate new optimal (fourth) order one step nodal cubic spline collocation methods for the solution of various elliptic boundary value problems in the unit square. These methods are constructed so that the respective collocation equations can be solved using matrix decomposition algorithms (MDAs). MDAs are fast, direct methods which employ fast Fourier transforms and require O(N 2 log N) operations on an N × N uniform partition of the unit square. The results of numerical experiments exhibit expected global optimal orders of convergence as well as desired superconvergence phenomena. © 2005 Society for Industrial and Applied Mathematics. 27 2 575 598 Cited By :7

Sponsors: ACM Special Interest Group on Computer Graphics and Interactive Techniques (SIGGRAPH) Conference code: 114475 Cited By :1

We consider the approximate solution of axisymmetric biharmonic problems using a boundary-type meshless method, the Method of Fundamental Solutions (MFS) with fixed singularities and boundary collocation. For such problems, the coefficient matrix of the linear system defining the approximate solution has a block circulant structure. This structure is exploited to formulate a matrix decomposition method employing fast Fourier transforms for the efficient solution of the system. The results of several numerical examples are presented. © Springer 2005. 23 1-2 55 71 Cited By :12

The development of the method of fundamental solutions (MFS) and related methods for the numerical solution of scattering and radiation problems in fluids and solids is described and reviewed. A brief review of the developments and applications in all areas of the MFS over the last five years is also given. Future possible areas of applications in fields related to scattering and radiation problems are identified. © 2003 Elsevier Ltd. All rights reserved. 27 7 759 769 Cited By :203

We consider the solution of various boundary value problems for the Helmholtz equation in the unit square using a nodal cubic spline collocation method and modifications of it which produce optimal (fourth-) order approximations. For the solution of the collocation equations, we formulate matrix decomposition algorithms, fast direct methods which employ fast Fourier transforms and require O(N2 log N) operations on an N × N uniform partition of the unit square. A computational study confirms the published analysis for the Dirichlet problem and indicates that similar results hold for Neumann, mixed, and periodic boundary conditions. The numerical results also exhibit superconvergence phenomena not reported in earlier studies. 24 5 1733 1753 Cited By :9

In this paper, we investigate the application of the Method of Fundamental Solutions (MFS) to two classes of axisymmetric potential problems. In the first, the boundary conditions as well as the domain of the problem, are axisymmetric, and in the second, the boundary conditions are arbitrary. In both cases, the fundamental solutions of the governing equations and their normal derivatives, which are required in the formulation of the MFS, can be expressed in terms of complete elliptic integrals. The method is tested on several axisymmetric problems from the literature and is also applied to an axisymmetric free boundary problem. Copyright © 1999 John Wiley & Sons, Ltd. 44 11 1653 1669 Cited By :48

The aim of this paper is to describe the development of the method of fundamental solutions (MFS) and related methods over the last three decades. Several applications of MFS-type methods are presented. Techniques by which such methods are extended to certain classes of non-trivial problems and adapted for the solution of inhomogeneous problems are also outlined. 9 1-2 69 95 Cited By :638