An iterative method for problems with multiscale conductivity
type: Item
instance of: scholarly article
title: An iterative method for problems with multiscale conductivity
zbMATH Open document ID: 1316.92044
DOI: 10.1155/2012/893040
author: Hyea Hyun Kim
author: Atul S. Minhas
author: Eung Je Woo
published in: Computational \& Mathematical Methods in Medicine
publication date: 2013-01-09T00:00:00Z
review text: Summary: A model with its conductivity varying highly across a very thin layer will be considered. It is related to a stable phantom model, which is invented to generate a certain apparent conductivity inside a region surrounded by a thin cylinder with holes. The thin cylinder is an insulator and both inside and outside the thin cylinderare filled with the same saline. The injected current can enter only through the holes adopted to the thin cylinder. The model has a high contrast of conductivity discontinuity across the thin cylinder and the thickness of the layer and the size of holes are very small compared to the domain of the model problem. Numerical methods for such a model require a very fine mesh near the thin layer to resolve the conductivity discontinuity. In this work, an efficient numerical method for such a model problem is proposed by employing a uniform mesh, which need not resolve the conductivity discontinuity. The discrete problem is then solved by an iterative method, where the solution is improved by solving a simple discrete problem with a uniform conductivity. At each iteration, the right-hand side is updated by integrating the previous iterate over the thin cylinder. This process results in a certain smoothing effect on microscopic structures and our discrete model can provide a more practical tool for simulating the apparent conductivity. The convergence of the iterative method is analyzed regarding the contrast in the conductivity and the relative thickness of the layer. In numerical experiments, solutions of our method are compared to reference solutions obtained from COMSOL, where very fine meshes are used to resolve the conductivity discontinuity in the model. Errors of the voltage in \(L^{2}\) norm follow \(O(h)\) asymptotically and the current density matches quitewell those from the reference solution for a sufficiently small mesh size \(h\). The experimental results present a promising feature of our approach for simulating the appare…
Mathematics Subject Classification ID: 92C55
Mathematics Subject Classification ID: 35Q92
zbMATH DE Number: 6123574
Wikidata QID: Q30459227
describes a project that uses: COMSOL
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.1155/2012/893040
OpenAlex ID: W2007697683
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DBLP publication ID: journals/cmmm/KimMW12
description: scientific article; zbMATH DE number 6123574