The role of the multiquadric shape parameters in solving elliptic partial differential equations

K e y w o r d s E l l i p t i c PDEs, Generalized multiquadries, Different shape parameters. 1. I N T R O D U C T I O N The interest in meshfree methods to solve PDEs has grown considerably in the past 15 years. The two principal reasons are: (1) mesh generation over twoand three-dimensional complicated domains may require weeks or months to produce a well-behaved mesh, and (2) the convergence rate of traditional methods are typically second order, requiring very fine discretization. The fine diseretization required may need more operations than meshfree methods, even though these traditional methods are compactly supported. The meshfree radial basis functions (RBFs) have been shown to be particularly attractive by Fedoseyev et al. [1] and Cheng et al. [2] because *Author to whom all correspondence should be addressed. 0898-1221/06/$ see front mat ter @ 2006 Elsevier Ltd. All rights reserved. Typeset by .42MS-TEX doi: 10.1016/j.camwa.2006.04.009 1336 J . WERTZ et al. of the exponential convergence of certain C °° RBFs that has been observed. Various RBFs have been successfully applied in a differential quadrature setting to obtain very accurate and efficient solutions to PDEs of engineering interest [3,4]. One of the most used RBFs is the multiquadric (MQ) RBF. The generalized MQ basis function, ~j(x), where x C ~d, is given by Cj(x) = [ ( x x j ) 2 +ca2]~. Commonly used values for/3 are 1 / 2 and 1/2. Madych and Nelson [5] and Madych [6] have proven theoretically tha t MQ interpolation converges exponentially as ~c/h, where r~ is a real number, r / < 1. Any continuous function, U(x), over the domain, f/, covered by a set of discrete points can be interpolated from the neighboring points of a point xi using RBFs and a polynomial basis as rz ~ U(x) = ~ ¢j(x)a~ + ~p0(x)~0, (1) i = 1 j = l with the constraint condition n E p j ( x i ) 5 ~ i O, j = 1 ,2 , . . . ,re, (2) i = 1 where ¢0(x) is the radial basis function, pj (x) is a monomial in the space coordinates x r = Ix, y], n is the number of data centers in f/, re is the number of polynomial basis functions (usually re < n), and a~ and l)j are the coefficients for Cj(x) and pj(x) , respectively, corresponding to the given point xi. The vectors are defined as a = [ < , a ~ , . . . ,an] ~ , (3) ~ T = [¢l (X) ,¢2(X), . . . ,¢n(X)] T , and ( 5 ) p T = } l ( x ) , p 2 ( x ) , . . . , p m ( x ) ] T (6) The radial distance function is a function of Euclidean distance r defined as v = V/x -xi) 2 -4(y -yi) 2 in two dimensions. (7) The radial distance function transforms a multiple-dimensional problem into a one-dimensional problem, and the polynomial term is added to ensure the conditional positive definiteness of the RBF approximation. The polynomial basis has following monomial terms: p m = [1 ,x ,y , x2 xy , y 2 , . . . ] . (s) Enforcing the interpolation to pass through all n scattered points within the influence domain leads to the following set of equations for the coefficients and 3' = [fi, l~] T and f.J = [U, 0] T, we solve the following equation: A3" = fd, (9)