Adaptive multiquadric collocation for boundary layer problems

AbstractAn adaptive collocation method based upon radial basis functions is presented for the solution of singularly perturbed two-point boundary value problems. Using a multiquadric integral formulation, the second derivative of the solution is approximated by multiquadric radial basis functions. This approach is combined with a coordinate stretching technique. The required variable transformation is accomplished by a conformal mapping, an iterated sine-transformation. A new error indicator function accurately captures the regions of the interval with insufficient resolution. This indicator is used to adaptively add data centres and collocation points. The method resolves extremely thin layers accurately with fairly few basis functions. The proposed adaptive scheme is very robust, and reaches high accuracy even when parameters in our coordinate stretching technique are not chosen optimally. The effectiveness of our new method is demonstrated on two examples with boundary layers, and one example featuring an interior layer. It is shown in detail how the adaptive method refines the resolution

Regularizing Ill-Posed Problems: Experiments With Multilevel Schemes

. A brief survey of regularization algorithms and parameter selection procedures relevant to multilevel schemes is provided. A number of new multilevel algorithms for solving ill-posed problems is introduced. They include a multilevel Landweber iteration that resembles the TwomeyTikhonov scheme, and a multilevel Tikhonov scheme with zero'th order and first-order stabilizers. It is shown that these algorithms with an appropriate discrepancy principle do possess the algebraic property of a standard multigrid method. The algorithms are applied to several linear and nonlinear first kind Fredholm integral equations. These numerical examples demonstrate significant speed-up (factor 3 to 8) for the new multilevel regularization techniques when compared to some standard regularization procedures. Key words. ill-posed equations, multilevel schemes, regularization algorithms, Morozov discrepancy principle, Landweber iteration, Twomey-Tikhonov iteration, Tikhonov regularization, stabilizers, con..

Boundary Layer Resolving Pseudospectral Methods For Singular Perturbation Problems

. Pseudospectral methods are investigated for singularly perturbed boundary value problems for ordinary differential equations which possess boundary layers. It is well known that if the boundary layer is very small then a very large number of spectral collocation points is required to obtain accurate solutions. We introduce here a new effective procedure, based on coordinate stretching and the Chebyshev pseudospectral method to resolve the boundary layers. Stable and accurate results are obtained for very thin boundary layers with a fairly small number of spectral collocation points. Key words. spectral methods, singular perturbation, boundary layer AMS subject classifications. 65N35 1. Introduction. We consider the pseudospectral (PS) method for the singular perturbation boundary value problem (BVP), given by fflu 00 (x) + p(x)u 0 (x) + q(x)u(x) = f(x); x 2 (\Gamma1; 1); u(\Gamma1) = ff; u(1) = fi; (1) where ffl ? 0 denotes a fixed (small) constant. In many applications, (1) p..

www.elsevier.com/locate/laa Persistently positive inverses of perturbed M-matrices

A well known property of an M-matrix A is that the inverse is element-wise non-negative, which we write as A −1 � 0. In this paper we consider perturbations of M-matrices and obtain bounds on the perturbations so that the non-negative inverse persists. The bounds are written in terms of decay estimates which characterize the decay (along rows) of the elements of the inverse matrix. We obtain results for diagonal and rank-1 perturbations of symmetric tridiagonal M-matrices and rank-1 perturbations of non-symmetric tridiagonal M-matrices. © 2007 Elsevier Inc. All rights reserved