A posteriori estimates for partial differential equations
type: Item
instance of: scholarly article
title: A posteriori estimates for partial differential equations
zbMATH Open document ID: 1162.65001
DOI: 10.1515/9783110203042
published in: Radon Series on Computational and Applied Mathematics
publication date: 2008-06-25T00:00:00Z
review text: This is an excellent textbook for a priori estimation for various mathematical problems described by partial differential equations. It can cover a number of numerical methods including the finite element method, although (or rather since) the employed methodology is in principle independent of specific approximation methods. It begins with the introductory chapter for some mathematical backgrounds in functional-analytical expression. Then it is followed by a concise overview of various a posteriori error estimation methods developed in the last century. The subsequent main chapters of the book are devoted to a new functional approach to a posteriori error control developed in the last decade. The main idea of this approach is as follows: an integral identity that defines a generalized solution is a source of guaranteed and computable error bounds between this solution and any function in the associated energy space. Those bounds can be derived by purely functional-analytic methods without relying on specific features of approximations or numerical methods. The basic ideas of this approach are presented by means of a simple elliptic problem in Chapter 3, and the following chapters are devoted to various important classes of problems, e.g., diffusion, linear elasticity, incompressible viscous fluids, nonlinear problems described with variational inequalities, evolutional problems. The text requires a moderate background in functional analysis and the theory of partial differential equations. It will be particularly useful for experts in computational mathematics, and also for students specializing in applied mathematics.
reviewed by: Fumio Kikuchi
Mathematics Subject Classification ID: 65-02
Mathematics Subject Classification ID: 65N15
Mathematics Subject Classification ID: 74S05
Mathematics Subject Classification ID: 76M10
Mathematics Subject Classification ID: 65N30
Mathematics Subject Classification ID: 74B05
Mathematics Subject Classification ID: 35J25
Mathematics Subject Classification ID: 76D05
Mathematics Subject Classification ID: 35Q30
Mathematics Subject Classification ID: 35J85
zbMATH DE Number: 5292566
zbMATH Keywords: a posteriori error estimates
zbMATH Keywords: numerical methods
zbMATH Keywords: textbook
zbMATH Keywords: finite element method
zbMATH Keywords: guaranteed and computable error bounds
zbMATH Keywords: elliptic problem
zbMATH Keywords: diffusion
zbMATH Keywords: linear elasticity
zbMATH Keywords: incompressible viscous fluids
zbMATH Keywords: variational inequalities
author: Sergey Repin
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.1515/9783110203042
OpenAlex ID: W408406386
description: scientific article; zbMATH DE number 5292566