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Operator Theory and Numerical Methods

  • 1st Edition, Volume 30 - July 3, 2001
  • Latest edition
  • Authors: H. Fujita, N. Saito, T. Suzuki
  • Language: English

In accordance with the developments in computation, theoretical studies on numerical schemes are now fruitful and highly needed. In 1991 an article on the finite element method… Read more

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Description

In accordance with the developments in computation, theoretical studies on numerical schemes are now fruitful and highly needed. In 1991 an article on the finite element method applied to evolutionary problems was published. Following the method, basically this book studies various schemes from operator theoretical points of view. Many parts are devoted to the finite element method, but other schemes and problems (charge simulation method, domain decomposition method, nonlinear problems, and so forth) are also discussed, motivated by the observation that practically useful schemes have fine mathematical structures and the converses are also true.

Readership

Students, R&D for experts, Engineers

Table of contents

Chapter 1. Elliptic Boundary Value Problems and FEM1.1 Elliptic Boundary Value Problems 1.2 Ritz-Galerkin Method 1.3 Finite Element Method (FEM) 1.4 Inverse Assumption 1.5 Loo Estimate 1.6 Lp Estimate 1.7 Asymptotic Expansion.Chapter 2. Semigroup Theory and FEM2.1 Evolutionary Problems 2.2 Semi-discretization 2.3 Fractional Powers 2.4 Full-discretization 2.5 Inhomogeneous Equation 2.6 Higher Accuracy 2.7 Loo Estimate 2.8 Hyperbolic Equation.Chapter 3. Evolution Equations and FEM3.1 Generation Theories3.2 A Priori Estimates 3.3 Semi-discretization 3.4 Full-discretization 3.5 Alternative Approach.Chapter 4. Other Methods in Time Discretization4.1 Rational Approximation of Semigroups4.2 Multi-step Method4.3 Product Formula.Chapter 5. Other Methods in Space Discretization5.1 Lumping of Mass5.2 Upwind Finite Elements5.3 Mixed Finite Elements5.4 Boundary Element Methods (BEM)5.5 Charge Simulation Methods (CSM).Chapter 6. Nonlinear Problems6.1 Semilinear Elliptic Equations6.2 Semilinear Parabolic Equations6.3 Degenerate Parabolic Equations.Chapter 7. Domain Decomposition Method7.1 Dirichlet to Neumann (DN) Map7.2 Dirichlet to Neumann (DN) Iteration7.3 Dirichlet2 to Neumann2 (DD-NN) Iteration7.4 Robin to Robin Iteration7.5 Exterior Problem7.6 The Stokes System.Bibliography. Index

Review quotes

"The authors provide a very sharp theoretical study of numerical methods used to solve partial diferential equations of elliptic and parabolic type. Every numerical scheme is thoroughly dissected. As a whole, everything fits together in a harmonious way."—Zentrallblatt fur Mathematik

"The book is efficiently organized and each chapter concludes with a very informative commentary section that provides brief but useful historical, bibliographical or technical comments. —Mathematical Reviews

Product details

  • Edition: 1
  • Latest edition
  • Volume: 30
  • Published: July 3, 2001
  • Language: English

About the authors

HF

H. Fujita

Affiliations and expertise
Tokai University, The Research Institute of Educational Development, Tokyo, Japan

NS

N. Saito

Affiliations and expertise
Toyama University, Faculty of Education, Toyama, Japan

TS

T. Suzuki

Affiliations and expertise
Osaka University, Department of Mathematics, Graduate School of Science, Toyonaka, Japan

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