Author(s)

D.-A. An-Vo, C.-D. Tran, N. Mai-Duy & T. Tran-Cong

Abstract

Many engineering problems have a wide range of length scales in their solutions. Direct numerical simulations for these problems typically require extremely-large amounts of CPU time and computer memory, which may be too expensive or impossible on the present supercomputers. In this paper, we present a high-order method, based on the multiscale basis function framework and integrated radialbasis- function networks, for solvingmultiscale elliptic problems in one dimension. Keywords: integrated radial basis functions, point collocation, subregion collocation, multiscale elliptic problems. 1 Introduction In composite materials, the presence of particles/fibres in the resin gives rise to the multiscale fluctuations in the thermal or electrical conductivity. In porous media, formation properties such as permeability have a very high degree of spatial variability. These effects are typically captured at scales that are too fine for direct numerical simulation. To enable the solution of these problems, a number of advanced numerical methods have been developed. Examples include those based on the homogenisation theory (e.g. [1]), upscaling methods (e.g. [2]) and multiscale methods (e.g. [3]). The homogenisation-theory-based methods have been successfully applied for the prediction of effective properties and statistical correlation lengths for multicomponent random media. However, restrictive assumptions on the media, such as scale separation and periodicity, limit their range of application. Furthermore, when dealing with problems having many separate scales, they become very expensive because their computational cost increases exponentially with the number of scales. For upscaling methods, their design

Keywords

integrated radial basis functions, point collocation, subregion collocation, multiscale elliptic problems