Weighted radial basis collocation method for boundary value problems

This work introduces the weighted radial basis collocation method for boundary value problems. We first show that the employment of least‐squares functional with quadrature rules constitutes an approximation of the direct collocation method. Standard radial basis collocation method, however, yields a larger solution error near boundaries. The residuals in the least‐squares functional associated with domain and boundary can be better balanced if the boundary collocation equations are properly weighted. The error analysis shows unbalanced errors between domain, Neumann boundary, and Dirichlet boundary least‐squares terms. A weighted least‐squares functional and the corresponding weighted radial basis collocation method are then proposed for correction of unbalanced errors. It is shown that the proposed method with properly selected weights significantly enhances the numerical solution accuracy and convergence rates. Copyright © 2006 John Wiley & Sons, Ltd.     

Direct solution of ill‐posed boundary value problems by radial basis function collocation method

Numerical solution of ill‐posed boundary value problems normally requires iterative procedures. In a typical solution, the ill‐posed problem is first converted to a well‐posed one by assuming the missing boundary values. The new problem is solved by a conventional numerical technique and the solution is checked against the unused data. The problem is solved iteratively using optimization schemes until convergence is achieved. The present paper offers a different procedure. Using the radial basis function collocation method, we demonstrate that the solution of certain ill‐posed problems can be accomplished without iteration. This method not only is efficient and accurate, but also circumvents the stability problem that can exist in the iterative method. Copyright © 2005 John Wiley & Sons, Ltd.     

Higher order Chebyshev basis functions for two-point boundary value problems

Cubic basis functions in one dimension for the solution of two-point boundary value problems are constructed based on the zeros of Chebyshev polynomials of the first kind. A general formula is derived for the construction of polynomial basis functions of degree r, where 1 ≤r < ∞. A Galerkin finite element method using the constructed basis functions for the cases r = 1, 2 and 3 is successfully applied to three different types of problem including a singular perturbation problem.     

A Cartesian‐grid collocation technique with integrated radial basis functions for mixed boundary value problems

In this paper, high‐order systems are reformulated as first‐order systems, which are then numerically solved by a collocation method. The collocation method is based on Cartesian discretization with 1D‐integrated radial basis function networks (1D‐IRBFN) (Numer. Meth. Partial Differential Equations 2007; 23 :1192–1210). The present method is enhanced by a new boundary interpolation technique based on 1D‐IRBFN, which is introduced to obtain variable approximation at irregular points in irregular domains. The proposed method is well suited to problems with mixed boundary conditions on both regular and irregular domains. The main results obtained are (a) the boundary conditions for the reformulated problem are of Dirichlet type only; (b) the integrated RBFN approximation avoids the well‐known reduction of convergence rate associated with differential formulations; (c) the primary variable (e.g. displacement, temperature) and the dual variable (e.g. stress, temperature gradient) have similar convergence order; (d) the volumetric locking effects associated with incompressible materials in solid mechanics are alleviated. Numerical experiments show that the proposed method achieves very good accuracy and high convergence rates. Copyright © 2009 John Wiley & Sons, Ltd.     

Testing complete and compact radial basis functions for solution of eigenvalue problems using the boundary element method with direct integration

This paper analyses the performance of the main radial basis functions in the formulation of the Boundary Element Method (DIBEM). This is an alternative for solving problems modeled by non-adjoint differential operators, since it transforms domain integrals in boundary integrals using radial basis functions. The solution of eigenvalue problem was chosen to performance evaluation. Natural frequencies are calculated numerically using several radial functions and their accuracy is evaluated by comparison with the available analytical solutions and with the Finite Element Method as well. The standard radial basis functions have presented similar performance to compact radial functions, being even slightly superior.     

On the increasingly flat radial basis function and optimal shape parameter for the solution of elliptic PDEs

For the interpolation of continuous functions and the solution of partial differential equation (PDE) by radial basis function (RBF) collocation, it has been observed that solution becomes increasingly more accurate as the shape of the RBF is flattened by the adjustment of a shape parameter. In the case of interpolation of continuous functions, it has been proven that in the limit of increasingly flat RBF, the interpolant reduces to Lagrangian polynomials. Does this limiting behavior implies that RBFs can perform no better than Lagrangian polynomials in the interpolation of a function, as well as in the solution of PDE? In this paper, arbitrary precision computation is used to test these and other conjectures. It is found that RBF in fact performs better than polynomials, as the optimal shape parameter exists not in the limit, but at a finite value.     

On the selection of coordinate functions for the solution of boundary problems by Galerkin's method

We propose a method for determining the coordinate functions to be employed in Galerkin's method for the solution of boundary problems, which increases significantly the precision of the calculations in a first approximation.Translated from Inzhenerno-Fizicheskii Zhurnal, Vol. 18, No. 2, pp. 309–315, February, 1970.     

A method for the numerical solution of a class of boundary value problems governed by second order elliptic systems

A method for the numerical solution of a class of problems governed by a system of second order elliptic partial differential equations is derived. The solution to the boundary value problem is obtained in terms of an integral taken round part of the boundary of the region under consideration. Some numerical examples are considered and the results obtained are shown to be in excellent agreement with those obtained either analytically or by employing other numerical methods.     

An iteration method using artificial boundary for some elliptic boundary value problems with singularities

By using an artificial boundary an iteration method is designed to solve some elliptic boundary value problems with singularities. At each step of the iteration the standard finite element method is used to solve the problems in a domain without singularities. It is shown that the iteration method is equivalent to a Schwarz alternating method. Some numerical examples are given, which show the effectiveness of the iteration method. Copyright © 1999 John Wiley & Sons, Ltd.     

Some comments on the use of radial basis functions in the dual reciprocity method

We comment on the use of radial basis functions in the dual reciprocity method (DRM), particularly thin plate splines as used in Agnantiaris, Polyzos and Beskos (1996). We note that the omission of the linear terms could have biased the numerical results as has occurred in several previous studies. Furthermore, we show that a full understanding of the convergence behavior of the DRM requires one to consider both interpolation and BEM errors, since the latter can offset the effect of improved data approximation. For a model Poisson problem this is demonstrated theoretically and the results confirmed by a numerical experiment.     

Numerical solution of fuzzy boundary value problems using Galerkin method

This paper proposes a new technique based on Galerkin method for solving nth order fuzzy boundary value problem. The proposed method has been illustrated by considering three different cases depending upon the sign of coefficients with benchmark example problems. To show the applicability of the proposed method, an application problem related to heat conduction has also been studied. The results obtained by the proposed methods are compared with the exact solution and other existing methods for demonstrating the validity and efficiency of the present method.     

Some recent results and proposals for the use of radial basis functions in the BEM

We survey some recent applications of radial basis functions (rbfs) for the BEM and related algorithms such as the method of fundamental solutions. Among these are the use of alternatives to the traditional 1+r function in the dual reciprocity method such as thin plate splines, multquadrics and the recently discovered compactly supported positive definite rbfs, and convergence proofs of the DRM for Poisson’s equation. Newly discovered particular solutions for Helmholtz-type operators are discussed and applied to give efficient mesh free algorithms for the diffusion equation. In addition, a number of proposals are given for future applications of rbfs such as the use of surface rbfs for interpolation and the solution of boundary integral equations and the application of Kansa’s method to develop new rbf based coupled domain-boundary approximation methods.     

Use of singularity capturing functions in the solution of problems with discontinuous boundary conditions

A method is proposed to improve the accuracy of the numerical solution of elliptic problems with discontinuous boundary conditions using both global and local meshless collocation methods with multiquadrics as basis functions. It is based on the use of special functions which capture the singular behavior near discontinuities in boundary conditions. In the case of global collocation, the method consists in enlarging the functional space spanned by the RBF basis functions, while in the case of local collocation, the method consists in modifying appropriately the problem in order to eliminate the singularities from the formulation. Numerical results for benchmark problems such as a stationary heat equation in a box (harmonic) and Stokes flow in a lid-driven square cavity, show significant improvements in accuracy and in compliance with the continuity equation.     

A numerical method for heat transfer problems using collocation and radial basis functions

Simple, mesh/grid free, numerical schemes for the solution of heat transfer problems are developed and validated. Unlike the mesh or grid-based methods, these schemes use well-distributed quasi-random collocation points and approximate the solution using radial basis functions. The schemes work in a similar fashion as finite differences but with random points instead of a regular grid system. This allows the computation of problems with complex-shaped boundaries in higher dimensions with no extra difficulty. © 1998 John Wiley & Sons, Ltd.     

Novel basis functions for the partition of unity boundary element method for Helmholtz problems

The BEM is a popular technique for wave scattering problems given its inherent ability to deal with infinite domains. In the last decade, the partition of unity BEM, in which the approximation space is enriched with a linear combination of plane waves, has been developed; this significantly reduces the number of DOFs required per wavelength. It has been shown that the element ends are more susceptible to errors in the approximation than the mid‐element regions. In this paper, the authors propose that this is due to the use of a collocation approach in combination with a reduced order of continuity in the Lagrangian shape function component of the basis functions. It is demonstrated, using numerical examples, that choosing trigonometric shape functions, rather than classical polynomial shape functions (quadratic in this case), provides accuracy benefits. Collocation schemes are investigated; it is found that the somewhat arbitrary choice of collocating at equally spaced points about the surface of a scatterer is better than schemes based on the roots of polynomials or consideration of the Fock domain. Copyright © 2012 John Wiley & Sons, Ltd.     

A dual-reciprocity boundary element method for a class of elliptic boundary value problems for non-homogeneous anisotropic media

A dual-reciprocity boundary element method is proposed for the numerical solution of a two-dimensional boundary value problem (BVP) governed by an elliptic partial differential equation with variable coefficients. The BVP under consideration has applications in a wide range of engineering problems of practical interest, such as in the calculation of antiplane stresses or temperature in non-homogeneous anisotropic media. The proposed numerical method is applied to solve specific test problems.     

Transformation of elliptic partial differential equations for solving two-dimensional boundary value problems in fluid flow

It is possible to transform elliptic partial differential equations to exchange the dependent with one of the independent variables. The Laplace equation for a stream function ‘Ψ’ over the X and Y co-ordinate system, for example, can be transformed into a relationship expressing the Y position of streamlines in terms of strem function Ψ and X. Although the resulting new partial differential equation is much more complex it is much more convenient to use in a computer. An irregular Y boundary becomes, with the new relationship, merely the boundary values assigned to the outer streamlines and the computer always need only deal with a rectangular array. The resulting answer is in the form of the position of streamlines which is the information directly required for plotting flow maps.