Michael A. Golberg

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

Abstract 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.

The method of fundamental solutions and quasi-Monte-Carlo method for diffusion equations

The Laplace transform is applied to remove the time-dependent variable in the diffusion equation. For non-harmonic initial conditions this gives rise to a non-homogeneous modified Helmholtz equation which we solve by the method of fundamental solutions. To do this a particular solution must be obtained which we find through a method suggested by Atkinson. To avoid costly Gaussian quadratures, we approximate the particular solution using quasi-Monte-Carlo integration which has the advantage of ignoring the singularity in the integrand. The approximate transformed solution is then inverted numerically using Stehfests algorithm. Two numerical examples are given to illustrate the simplicity and effectiveness of our approach to solving diffusion equations in 2-D and 3-D.

BEM for theomoelasticity and elasticity with body force — a revisit

For thermoelasticity and elasticity with a body force, the regular boundary element method involves a domain integral. Special techniques have been devised to eliminate the domain integral by either exact treatment, or approximation using radial basis functions. This paper gives a unified presentation of these techniques. Body forces of gravity, centrifugal, gradient, harmonic, and arbitrary types are investigated. In the approximation by radial basis function, existing work is extended to include higher order basis functions of conical, spline, and polynomial types.

The convergence of several algorithms for solving integral equations with finite part integrals. II

We establish the uniform convergence and obtain convergence rates for several algorithms for solving a class of Hadamard singular integral equations. These results improve on the mean square convergence shown in [2].

Compactly supported radial basis functions for shallow water equations

This paper presents the application of the compactly supported radial basis functions (CSRBFs) in solving a system of shallow water hydrodynamics equations. The proposed scheme is derived from the idea of piecewise polynomial interpolation using a function of Euclidean distance. The compactly supported basis functions consist of a polynomial which are non-zero on [0,1) and vanish on [1,~). This reduces the original resultant full matrix to a sparse matrix. The operation of the banded matrix system could reduce the ill-conditioning of the resultant coefficient matrix due to the use of the global radial basis functions. To illustrate the computational efficiency and accuracy of the method, the difference between the globally and CSRBF schemes is compared. The resulting banded matrix has shown improvement in both ill-conditioning and computational efficiency. The numerical solutions are verified with the observed data. Excellent agreement is shown between the simulated and the observed data.

Numerical solution of a class of integral equations arising in two-dimensional aerodynamics

We consider the numerical solution of a class of integral equations arising in the determination of the compressible flow about a thin airfoil in a ventilated wind tunnel. The integral equations are of the first kind with kernels having a Cauchy singularity. Using appropriately chosen Hilbert spaces, it is shown that the kernel gives rise to a mapping which is the sum of a unitary operator and a compact operator. This enables us to study the problem in terms of an equivalent integral equation of the second kind. Using Galerkins method, we are able to derive a convergent numerical algorithm for its solution. It is shown that this algorithm is numerically equivalent to Blands collocation method, which is then used as our method of computation. Extensive numerical calculations are presented establishing the validity of the theory.

The convergence of a collocation method for a class of Cauchy singular integral equations

In several recent papers we have studied the numerical solution of Cauchy singular integral equations of the first kind by a polynomial collocation method [4-61. This technique has proved to be quite efftcient for many problems, even when the kernel (but not the right-hand side) is discontinuous [4-61. The L, convergence and stability was established in [S] and [6]. In [ 121 Ioakimidis extended this method to solve Cauchy singular integral equations of the second kind with constant coefficients. Although some discussion of convergence was given it appears (at least to this author) that the proof is incomplete and the conditions on the kernel and the right-hand side are somewhat more restrictive than necessary. In this paper we will show that the convergence analysis given in [5,6] can be generalized to establish the L, convergence of collocation under rather mild restrictions on the data. Although a variety of numerical techniques such as Galerkin’s method and direct quadrature have been used extensively to solve Cauchy singular equations, they seem to be generally suited for problems with smooth kernels. For problems with discontinuous kernels collocation methods appear to be a reasonable compromise between time-consuming Galerkin methods [2] and direct quadrature methods which require the evaluation of the kernel at pairs of points where the kernel may become unbounded [3, 131. It is well known that for Fredholm equations collocation is related to product integration quadrature methods [ 11. Since such methods seem not to have been developed for the solution of Cauchy singular equations collocation appears to be a good way of dealing with fairly general integral equations of this type [4, 71.

On the L2 convergence of collocation for the generalized airfoil equation

Abstract In this paper we establish the L2 convergence of a polynomial collocation method for the solution of a class of Cauchy singular linear integral equations, which we term the generalized airfoil equation. Previous numerical results have shown that if the right hand side is smooth then convergence is rapid, with 6 decimal accuracy achievable using 8–10 basis elements. Practical problems in aerodynamics dictate that this equation be solved for discontinuous data. The convergence rate is numerically demonstrated to be O( 1 N ) , where N is the number of basis elements used. Simple extrapolation is shown to be effective in accelerating the convergence, 4–5 decimal accuracy being achieved using 16 basis elements.

On a method of Atkinson for evaluating domain integrals in the boundary element method

We have shown how to couple a numerical method due to Atkinson for computing particular solutions to a class of elliptic differential equations with a variety of boundary element methods, which alleviates the problem of domain discretization for solving inhomogeneous equations. When the inhomogeneous term is of the form typically used in the dual reciprocity method, Atkinsons formula is such that only boundary integrals need to be approximated numerically. If this method is coupled with the method of potentials, it results in a computational technique which requires neither boundary nor domain discretization. Some numerical results are given validating our approach.

The annihilator method for computing particular solutions to partial differential equations

Abstract We generalize the well-known annihilator method, used to find particular solutions for ordinary differential equations, to partial differential equations. This method is then used to find particular solutions of Helmholtz-type equations when the right hand side is a linear combination of thin plate and higher order splines. These particular solutions are useful in numerical algorithms for solving boundary value problems for a variety of elliptic and parabolic partial differential equations.