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Least-Squares Finite Element Methods

  • Book
  • © 2009

Overview

Authors:
  1. Max D. Gunzburger
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    You can also search for this author in PubMed   Google Scholar

  2. Pavel B. Bochev
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  • Puts least-squares finite element methods on a common mathematically sound foundation
  • Reviews strengths and weaknesses, successes and open problems of finite element methods
  • Appendices include results from functional analysis and standard finite theory
  • Includes supplementary material: sn.pub/extras

Part of the book series: Applied Mathematical Sciences (AMS, volume 166)

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Table of contents (16 chapters)

  1. Front Matter

    Pages 1-21
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  2. Survey of Variational Principles and Associated Finite Element Methods.

    1. Front Matter

      Pages 1-1
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    2. Classical Variational Methods

      • Pavel B. Bochev, Max D. Gunzburger
      Pages 1-31

    3. Alternative Variational Formulations

      • Pavel B. Bochev, Max D. Gunzburger
      Pages 1-31

  3. Abstract Theory of Least-Squares Finite Element Methods

    1. Front Matter

      Pages 1-1
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    2. Mathematical Foundations of Least-Squares Finite Element Methods

      • Pavel B. Bochev, Max D. Gunzburger
      Pages 1-33

    3. The Agmon–Douglis–Nirenberg Setting for Least-Squares Finite Element Methods

      • Pavel B. Bochev, Max D. Gunzburger
      Pages 1-28

  4. Least-Squares Finite Element Methods for Elliptic Problems

    1. Front Matter

      Pages 1-1
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    2. Scalar Elliptic Equations

      • Pavel B. Bochev, Max D. Gunzburger
      Pages 1-64

    3. Vector Elliptic Equations

      • Pavel B. Bochev, Max D. Gunzburger
      Pages 1-40

    4. The Stokes Equations

      • Pavel B. Bochev, Max D. Gunzburger
      Pages 1-72

  5. Least-Squares Finite Element Methods for Other Settings

    1. Front Matter

      Pages 1-1
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    2. The Navier–Stokes Equations

      • Pavel B. Bochev, Max D. Gunzburger
      Pages 1-55

    3. Parabolic Partial Differential Equations

      • Pavel B. Bochev, Max D. Gunzburger
      Pages 1-36

    4. Hyperbolic Partial Differential Equations

      • Pavel B. Bochev, Max D. Gunzburger
      Pages 1-26

    5. Control and Optimization Problems

      • Pavel B. Bochev, Max D. Gunzburger
      Pages 1-46

    6. Variations on Least-Squares Finite Element Methods

      • Pavel B. Bochev, Max D. Gunzburger
      Pages 1-56

  6. Supplementary Material

    1. Front Matter

      Pages 1-1
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    2. Analysis Tools

      • Pavel B. Bochev, Max D. Gunzburger
      Pages 1-19

    3. Compatible Finite Element Spaces

      • Pavel B. Bochev, Max D. Gunzburger
      Pages 1-32
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Keywords

About this book

Since their emergence in the early 1950s, ?nite element methods have become one of the most versatile and powerful methodologies for the approximate numerical solution of partial differential equations. At the time of their inception, ?nite e- ment methods were viewed primarily as a tool for solving problems in structural analysis. However, it did not take long to discover that ?nite element methods could be applied with equal success to problems in other engineering and scienti?c ?elds. Today, ?nite element methods are also in common use, and indeed are often the method of choice, for incompressible ?uid ?ow, heat transfer, electromagnetics, and advection-diffusion-reaction problems, just to name a few. Given the early conn- tion between ?nite element methods and problems engendered by energy minimi- tion principles, it is not surprising that the ?rst mathematical analyses of ?nite e- ment methods were given in the environment of the classical Rayleigh–Ritz setting. Yet again, using the fertile soil provided by functional analysis in Hilbert spaces, it did not take long for the rigorous analysis of ?nite element methods to be extended to many other settings. Today, ?nite element methods are unsurpassed with respect to their level of theoretical maturity.

Reviews

From the reviews: “In the book under review, the authors give a unified and comprehensive treatment of least-squares finite element methods and discuss important implementation issues that are critical to their success in practice. … This book is valuable both for researchers and practitioners working in least-squares finite element methods. … In addition, others will find it a great reference for learning about the theory and implementation of the least-squares finite element methods.” (Tsu-Fen Chen, Mathematical Reviews, Issue 2010 b)

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