This monograph presents a summary account of the subject of a posteriori error estimation for finite element approximations of problems in mechanics. The study primarily focuses on methods for linear elliptic boundary value problems. However, error estimation for unsymmetrical systems, nonlinear problems, including the Navier-Stokes equations, and indefinite problems, such as represented by the Stokes problem are included. The main thrust is to obtain error estimators for the error measured in the energy norm, but techniques for other norms are also discussed.

A Posteriori Error Estimation in Finite Element Analysis

This book is a tutorial written by researchers and developers behind the FEniCS Project and explores an advanced, expressive approach to the development of mathematical software. The presentation spans mathematical background, software design and the use of FEniCS in applications. Theoretical aspects are complemented with computer code which is available as free/open source software. The book begins with a special introductory tutorial for beginners. Followingare chapters in Part I addressing fundamental aspects of the approach to automating the creation of finite element solvers. Chapters in Part II address the design and implementation of the FEnicS software. Chapters in Part III present the application of FEniCS to a wide range of applications, including fluid flow, solid mechanics, electromagnetics and geophysics.

Automated Solution of Differential Equations by the Finite Element Method: The FEniCS Book

Introduction. A Simple Model Problem. Abstract Nonlinear Equations. Finite Element Discretizations of Elliptic PDEs. Practical Implementation. Bibliography. Subject Index.

A Review of a Posteriori Error Estimation and Adaptive Mesh-Refinement Techniques

Advances in scientific computing have made modelling and simulation an important part of the decision-making process in engineering, science, and public policy. This book provides a comprehensive and systematic development of the basic concepts, principles, and procedures for verification and validation of models and simulations. The emphasis is placed on models that are described by partial differential and integral equations and the simulations that result from their numerical solution. The methods described can be applied to a wide range of technical fields, from the physical sciences, engineering and technology and industry, through to environmental regulations and safety, product and plant safety, financial investing, and governmental regulations. This book will be genuinely welcomed by researchers, practitioners, and decision makers in a broad range of fields, who seek to improve the credibility and reliability of simulation results. It will also be appropriate either for university courses or for independent study.

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Verification and Validation in Scientific Computing

We construct a converging adaptive algorithm for linear elements applied to Poisson’s equation in two space dimensions. Starting from a macro triangulation, we describe how to construct an initial triangulation from a priori information. Then we use a posteriors error estimators to get a sequence of refined triangulation and approximate solutions. It is proved that the error, measured in the energy norm, decreases at a constant rate in each step until a prescribed error bound is reached. Extension to higher-order elements in two space dimension and numerical results are included.

A convergent adaptive algorithm for Poisson's equation

We present three new a posteriori error estimators in the energy norm for finite element solutions to elliptic partial differential equations. The estimators are based on solving local Neumann problems in each element. The estimators differ in how they enforce consistency of the Neumann problems. We prove that as the mesh size decreases, under suitable assumptions, two of the error estimators approach upper bounds on the norm of the true error, and all three error estimators are within multiplicative constants of the norm of the true error. We present numerical results in which one of the error estimators appears to converge to the norm of the true error.

/pdf/some-a-posteriori-error-estimators-for-elliptic-partial-3u6sm3nup8.pdf

Some a posteriori error estimators for elliptic partial differential equations

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Some a posteriori error estimators for elliptic partial differential equations