A matrix decomposition MFS algorithm for axisymmetric biharmonic problems

Fairweather, Graeme; Karageorghis, Andréas; Smyrlis, Yiorgos–Sokratis

doi:10.1007/s10444-004-1808-6

Cited by 15 publications

(9 citation statements)

References 20 publications

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“…Some preliminary experiments for a three-dimensional geometry we carried out and indicate that conclusions analogous to the ones reported for the twodimensional cases are also valid in three dimensions. The extension of the new approach and its efficient implementation in the case of axisymmetric domains (see, [27]) and in the case of different differential operators (see, [9]) will be the subject of a different study. Proof.…”

Section: The Ill-conditioning Of the Mfs Matricesmentioning

confidence: 99%

The under-determined version of the MFS: Taking more sources than collocation points

Smyrlis

Karageorghis

2010

Applied Numerical Mathematics

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Abstract. In this study we investigate the approximation of the solutions of certain elliptic boundary value problems by the Method of Fundamental Solutions (MFS). In particular, we study the case in which the number of singularities (sources) exceeds the number of boundary (collocation) points and we propose two methods of determining which approximate solution is optimal. We also develop an efficient numerical algorithm for the Dirichlet problem for Laplace's equation in the special case the domain of the problem is a disk. Numerical experiments are presented which reveal that in the case where the number of singularities is twice the number of boundary points, the conditioning of the MFS matrices and the concomitant accuracy of the method improve significantly.

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Section: The Ill-conditioning Of the Mfs Matricesmentioning

confidence: 99%

The under-determined version of the MFS: Taking more sources than collocation points

Smyrlis

Karageorghis

2010

Applied Numerical Mathematics

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“…MDAs have also been used in applications of the method of fundamental solutions (MFS) for the solution of certain harmonic and biharmonic axisymmetric problems [92,181,182,191] and axisymmetric problems in linear elasticity and thermoelasticity [126,127]. As an example, we describe the MFS MDA proposed in [181] for the three-dimensional axisymmetric potential problem,…”

Section: The Methods Of Fundamental Solutions and Related Techniquesmentioning

confidence: 98%

Matrix decomposition algorithms for elliptic boundary value problems: a survey

2010

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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 log N) 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

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“…, i, j = 1, 2, 3, of the coefficient matrix H in (3.4) has a block circulant structure [8,22] of the form…”

Section: Matrix Decomposition Algorithm Each Of the Nine Blocks H Ijmentioning

confidence: 99%

“…Because of the block circulant structure of the matrices H ij , following a technique similar to the ones employed in [8,22], for every i, j = 1, 2, 3, we have that…”

Section: Matrix Decomposition Algorithm Each Of the Nine Blocks H Ijmentioning

confidence: 99%

“…In the current work, in both the two-and the three-dimensional cases, the MFS discretization leads to the solution of linear systems whose coefficient matrices have block circulant structures which can be exploited by using fast Fourier transforms (FFTs). The circulant and/or block circulant structure of the matrices appearing when the MFS is applied to axisymmetric problems has been exploited in [8,22]. The idea of exploiting the FFTs to solve axisymmetric integral equations was introduced in [12]; see also [16].…”

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

The Method of Fundamental Solutions for Stationary Heat Conduction Problems in Rotationally Symmetric Domains

Smyrlis¹,

Karageorghis²

2006

SIAM J. Sci. Comput.

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Abstract. We propose an efficient boundary collocation method for the solution of certain two-and three-dimensional problems of steady-state heat conduction in isotropic bimaterials. In particular, in two dimensions we consider the case where a circular region composed of one material is coated with an annular region of another material. In three dimensions, we examine the corresponding case for axisymmetric domains. The proposed method involves the use of a domain decomposition technique in conjunction with a matrix decomposition algorithm. The circulant structure of the matrices appearing in this method is exploited by using fast Fourier transforms. The method is tested numerically on several problems.

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A matrix decomposition MFS algorithm for axisymmetric biharmonic problems

Cited by 15 publications

References 20 publications

The under-determined version of the MFS: Taking more sources than collocation points

The under-determined version of the MFS: Taking more sources than collocation points

Matrix decomposition algorithms for elliptic boundary value problems: a survey

The Method of Fundamental Solutions for Stationary Heat Conduction Problems in Rotationally Symmetric Domains

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