Least-squares finite element methods
type: Item
instance of: scholarly article
title: Least-squares finite element methods
zbMATH Open document ID: 1168.65067
author: Pavel B. Bochev
author: Max Gunzburger
published in: Applied Mathematical Sciences
publication date: 2008-06-25T00:00:00Z
review text: Part I with 2 chapters provides a brief review of the finite element method and an introduction into the least-squares method. The latter can deal only with algebraic properties of the method, since the method associates to the abstract equation \(Lu=f\) with \(L:X \to Y\) the least-squares problem \[ \|Lu-f\|_0^2 \to \min! \] and the (Ritz-Galerkin) characterization of the minimum corresponds to the equation \(L^*Lu=L^*f\). Thus the condition number is squared, but this is not wanted. Therefore, Part II discusses concepts, how appropriate least-squares formulations for partial differential equations are obtained. It is easily described for the Poisson equation \(\Delta\phi = f\) which is rewritten as a system of first order: \(\nabla\phi = v\) and \(\text{div } v = f\). Associate to the system the problem \[ \|\nabla\phi - v \|_0^2 + \|\text{div } v - f\|_s^2 \to \min! \] where the norms refer to the Sobolev norms with the associated index. The discretization of the least-squares problem by finite elements yields only the optimal order of the error if the index \(s\) above is chosen such that system of partial differential equation induces an isomorphism between the Sobolev spaces. If this is true, one achieves \textit{energy balance} in terms of the book. Chapter 4 in Part II and Part III describe how the Sobolev spaces with energy balance are determined from the Agmon-Douglis-Nirenberg setting. We note that the theory by Babuška and Brezzi is applied in the finite element method for similar reasons. The most practical choice for the Poisson equation is obviously \(s=0\), but the energy balance requires \(s=-1\), i.e., a negative norm. Fortunately, the negative norm can be replaced by mesh-dependent norms, and it is elucidated later by numerical results that the mesh-dependent weights are essential. In Part III also the least-squares methods for several formulations of diffusion problems, electromagnetics and the Stokes problem are included. Some of them are proba…
reviewed by: D. Braess
Mathematics Subject Classification ID: 65N30
Mathematics Subject Classification ID: 35Q60
Mathematics Subject Classification ID: 35K15
Mathematics Subject Classification ID: 35L15
Mathematics Subject Classification ID: 65M55
Mathematics Subject Classification ID: 65N55
Mathematics Subject Classification ID: 65M60
Mathematics Subject Classification ID: 35J05
Mathematics Subject Classification ID: 46E35
Mathematics Subject Classification ID: 35Q30
zbMATH DE Number: 5292600
zbMATH Keywords: least-squares method
zbMATH Keywords: discrete DeRham complex
zbMATH Keywords: finite element method
zbMATH Keywords: condition number
zbMATH Keywords: Poisson equation
zbMATH Keywords: energy balance
zbMATH Keywords: Sobolev spaces
zbMATH Keywords: numerical results
zbMATH Keywords: diffusion problems
zbMATH Keywords: electromagnetics
zbMATH Keywords: Stokes problem
zbMATH Keywords: Navier-Stokes equations
zbMATH Keywords: parabolic
zbMATH Keywords: hyperbolic
zbMATH Keywords: advection-diffusion problems
zbMATH Keywords: higher order elements
zbMATH Keywords: domain decomposition
MaRDI profile type: Publication
full work available at URL: b13382
OpenAlex ID: W1555557982
DOI: 10.1007/B13382
description: scientific article; zbMATH DE number 5292600