Domain Decomposition and Discretization of Continuous Mathematical Models
An International Journal computers & mathematics with applications PERGAMON Computers and Mathematics with Applications 38 (1999) 79-88 www.elsevier.nl/locate/camwa Domain Decomposition and Discretization of Continuous Mathematical Models I. BONZANI Department of Mathematics, Potitecnico, Torino, Italy Bonzani©Polito. it (Received April 1999; accepted May 1999) Abstract--This paper deals with the development of decomposition of domains methods related to the discretization, by collocation-interpolation methods, of continuous models described by nonlinear partial differential equations. The objective of this paper is to show how generalized collocation and domain decomposition methods can be applied for problems where different models are used within the same domain. © 1999 Elsevier Science Ltd. All rights reserved. Keywords--Nonlinear sciences, Evolution equations, Sinc functions, Domain decomposition, Dif- ferential quadrature. 1. INTRODUCTION A standard solution technique of nonlinear initial-boundary value problem for partial differential equations is the generalized collocation-interpolation method, originally proposed as differential quadrature method. This method, as documented in [1], as well as in some more recent devel- opments [2,3], discretizes the original continuous model (and problem) into a discrete (in space) model, with a finite number of degrees of freedom, while the initial-boundary value problem is transformed into an initial-value problem for ordinary differential equations. This method is well documented in the literature on applied mathematics. It was proposed by Bellman and Casti [1] and developed by several authors in the deterministic and stochastic framework as documented in Chapters 3 and 4 of [4]. The validity of the method, with respect to alternative ones, e.g., finite elements and volumes [5,6], Trefftz methods [7,8], spectral meth- ods [9,10], is not here discussed.
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