A comparison of efficiency and error convergence of multiquadric collocation method and finite element method

Li, Jichun; Cheng, Alexander H.‐D.; Chen, Ching-Shyang

doi:10.1016/s0955-7997(02)00081-4

Cited by 104 publications

(48 citation statements)

References 25 publications

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“…These traditional numerical methods require the data to be arranged in a structured pattern and to be contained in a simply shaped region, and, therefore, these approaches does not suit multidimensional practical problems. The Radial basis function (RBF) method is viewed as an important alternative numerical technique for multidimensional problems because it contains more beneficial properties in high-order accuracy and flexibility on geometry of the computational domain than classical techniques [1][2][3]. The main idea of the RBF method is approximating the solution in terms of linear combination of infinitely differentiable RBF φ, which is the function that depends only on the distance to a center point x and shape parameter ε.…”

Section: Introductionmentioning

confidence: 99%

“…Similar to the finite difference approach, the key idea is approximating the differential operator of solutions at each interior node by using a linear combination of the function values at the neighboring node locations and then determining the FD-weights so that the approximations become exact for all the RBFs that are centered at the neighboring nodes. Straightforward algebra will then show that the FD-weights can be obtained by solving the linear system, for which the system matrix is the same as matrix A in Equation (2). The invertibility of A also implies that FD-weights can always be computed.…”

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

An Investigation of Radial Basis Function-Finite Difference (RBF-FD) Method for Numerical Solution of Elliptic Partial Differential Equations

Yensiri

Skulkhu

2017

Mathematics

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Abstract:The Radial Basis Function (RBF) method has been considered an important meshfree tool for numerical solutions of Partial Differential Equations (PDEs). For various situations, RBF with infinitely differentiable functions can provide accurate results and more flexibility in the geometry of computation domains than traditional methods such as finite difference and finite element methods. However, RBF does not suit large scale problems, and, therefore, a combination of RBF and the finite difference (RBF-FD) method was proposed because of its own strengths not only on feasibility and computational cost, but also on solution accuracy. In this study, we try the RBF-FD method on elliptic PDEs and study the effect of it on such equations with different shape parameters. Most importantly, we study the solution accuracy after additional ghost node strategy, preconditioning strategy, regularization strategy, and floating point arithmetic strategy. We have found more satisfactory accurate solutions in most situations than those from global RBF, except in the preconditioning and regularization strategies.

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Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

An Investigation of Radial Basis Function-Finite Difference (RBF-FD) Method for Numerical Solution of Elliptic Partial Differential Equations

Yensiri

Skulkhu

2017

Mathematics

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“…Since then, DRBF methods have been increasingly used for the 10 solution of elliptic, parabolic and hyperbolic PDEs which govern many engineering problems. In [7,8,9,10,11], practitioners demonstrated that the elliptic PDE solutions using DRBFs converge much faster than those based on polynomial approximations. Mai-Duy and Tran-Cong proposed the idea of using Indirect/Integrated RBFs (IRBFs) for the solution of PDEs [12,13].…”

Section: Introductionmentioning

confidence: 99%

“…In this work, it is denoted by β. Numerical experiments indicated that the optimal value of β depends on the function to be interpolated, the configuration of nodal points, the RBF type, and the machine precision [3,4,9,20,21,22,23]. The matrix condi-30 tion of the RBF method grows exponentially with the RBF width.…”

Section: Introductionmentioning

confidence: 99%

A numerical study of compact approximations based on flat integrated radial basis functions for second-order differential equations

Tien

Mai‐Duy

Tran

et al. 2016

Computers & Mathematics with Applications

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In this paper, we propose a simple but effective preconditioning technique to improve the numerical stability of Integrated Radial Basis Function (IRBF) methods. The proposed preconditioner is simply the inverse of a well-conditioned matrix that is constructed using non-flat IRBFs. Much larger values of the free shape parameter of IRBFs can thus be employed and better accuracy for smooth solution problems can be achieved. Furthermore, to improve the accuracy of local IRBF methods, we propose a new stencil, namely Combined Compact IRBF (CCIRBF), in which (i) the starting point is the fourth-order derivative; and (ii) nodal values of first-and second-order derivatives at side nodes of the stencil are included in the computation of first-and second-order derivatives at the middle node in a natural way. The proposed stencil can be employed in uniform/nonuniform Cartesian grids. The preconditioning technique in combination with the CCIRBF scheme employed with large values of the shape parameter are tested with elliptic equations and then applied to simulate several fluid flow problems governed by Poisson, Burgers, convectiondiffusion, and Navier-Stokes equations. Highly accurate and stable solutions are obtained. In some cases, the preconditioned schemes are shown to be several orders of magnitude more accurate than those without preconditioning.

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“…It is a strong-form meshless method and unlike the weak-form meshless methods it is a truly meshless one since there is no integration in the solution procedure. There is numerical evidence that the method has an exponential convergence rate [6] and it is shown to be more accurate then the finite difference, finite element with linear shape functions and spectral methods [7,8]. On the other hand the method is less stable then weak-form meshless or mesh-based methods, but the high level of accuracies that can be obtained by fewer nodes motivates the use of this approximation scheme.…”

Section: Introductionmentioning

confidence: 99%

Numerical Solution of the Multigroup Neutron Diffusion Equation by the Meshless RBF Collocation Method

Tanbay

Ozgener

2013

MCA

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Abstract-The multigroup neutron diffusion criticality problem is studied by the radial basis function collocation method. The multiquadric is chosen as the radial basis function. To investigate the effectiveness of the method, one, two and three-group problems are considered. It is found that the radial basis function collocation method produces highly accurate multiplication factors and it is also efficient in the calculation of group fluxes.

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A comparison of efficiency and error convergence of multiquadric collocation method and finite element method

Cited by 104 publications

References 25 publications

An Investigation of Radial Basis Function-Finite Difference (RBF-FD) Method for Numerical Solution of Elliptic Partial Differential Equations

An Investigation of Radial Basis Function-Finite Difference (RBF-FD) Method for Numerical Solution of Elliptic Partial Differential Equations

A numerical study of compact approximations based on flat integrated radial basis functions for second-order differential equations

Numerical Solution of the Multigroup Neutron Diffusion Equation by the Meshless RBF Collocation Method

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