A finite element method for singularly perturbed reaction-diffusion problems.
type: Item
instance of: scholarly article
title: A finite element method for singularly perturbed reaction-diffusion problems.
zbMATH Open document ID: 1033.65101
author: Dali Zhang
author: Huo-Yuan Duan
published in: Acta Mathematicae Applicatae Sinica. English Series
publication date: 2004-03-19T00:00:00Z
review text: The authors present a stabilized FOLS method for solving the reaction-diffusion equation \(-\kappa\Delta u+\sigma u=f\) for small \(\kappa\) and large \(\sigma\). This equation can be expressed as a system of equations for \(u\) and \(p=-\nabla u\), and then written in terms of the least-squares operator \[ B(u,p;v,q)=\kappa\sigma(p+\nabla u,q+\nabla v)_0+(\kappa\nabla\cdot p+\sigma u,\kappa\nabla\cdot q+\sigma u)_0 \] and a stabilization term \[ T(u,p,v,q)=-\alpha\sum_K h_K^2(\sigma u+\kappa\nabla\cdot p,\sigma v+\kappa\nabla\cdot q)_{0,K}/(\alpha h_K^2+\kappa), \] where the sum is taken over the set of all elements \(K\), each of which has characteristic size \(h_K\). The authors prove that the discretized operator is coercive and that solutions of the discrete system converge. The norm used depends in part on the stabilization term. Furthermore, the condition number of the discrete stiffness matrix is \(O(1/h^2)\), uniformly for \(\sigma\geq\kappa\).
Mathematics Subject Classification ID: 65N30
Mathematics Subject Classification ID: 65N12
Mathematics Subject Classification ID: 35J25
Mathematics Subject Classification ID: 35B25
zbMATH DE Number: 1916589
zbMATH Keywords: singular perturbation
zbMATH Keywords: finite elements
zbMATH Keywords: FOLS method
zbMATH Keywords: reaction-diffusion equation
zbMATH Keywords: least-squares
zbMATH Keywords: stabilization
zbMATH Keywords: condition number
reviewed by: Myron Sussman
MaRDI profile type: Publication
full work available at URL: s10255-003-0077-3
OpenAlex ID: W2022286143
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DOI: 10.1007/S10255-003-0077-3
description: scientific article; zbMATH DE number 1916589