Efficient Kansa-type MFS algorithm for elliptic problems

Karageorghis, Andréas

doi:10.1007/s11075-009-9334-8

Cited by 10 publications

(4 citation statements)

References 16 publications

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“…This method avoids finding the optimal positions for sources in the MFS when the two-stage scheme is used. In contrast, another meshless method, which was proposed recently [15][16][17], uses only the singular fundamental solutions to approximate the solution of the differential equation. In such an approach, the mathematical derivation of the corresponding particular solution can be avoided.…”

Section: The Methods Of Approximate Particular Solutionsmentioning

confidence: 99%

The method of approximate particular solutions for solving certain partial differential equations

Chen

Fan

Wen

2010

Numerical Methods Partial

142

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A standard approach for solving linear partial differential equations is to split the solution into a homogeneous solution and a particular solution. Motivated by the method of fundamental solutions for solving homogeneous equations, we propose a similar approach using the method of approximate particular solutions for solving linear inhomogeneous differential equations without the need of finding the homogeneous solution. This leads to a much simpler numerical scheme with similar accuracy to the traditional approach. To demonstrate the simplicity of the new approach, three numerical examples are given with excellent results.

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Section: The Methods Of Approximate Particular Solutionsmentioning

confidence: 99%

The method of approximate particular solutions for solving certain partial differential equations

Chen

Fan

Wen

2010

Numerical Methods Partial

142

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“…Construct B m = Systems possessing block circulant structures also arise in the application of the MFS to harmonic and biharmonic problems in regular polygonal domains [122], and in the application of the so-called MFS-K to problems in circular domains [123]. MFS MDAs have also been developed for wave scattering by circular cylinders [5,192,193].…”

Section: Mfs Mdamentioning

confidence: 99%

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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“…Such basis functions are particularly appropriate for the collocation of Helmholtz type PDEs since they simplify the evaluation of the Laplace operator and differentiation is avoided. An efficient algorithm for solving the MFS-K collocation linear system for BVPs in axisymmetric domains was presented in [20].…”

Section: Introductionmentioning

confidence: 99%