Domain Decomposition Methods for Partial Differential Equations

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ISBN: 9780198501787 | 0198501781
Cover: Hardcover
Copyright: 7/29/1999

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A relatively new field, domain composition methods draw on parallel computing techniques and are proving a powerful approach to the numerical solution of partial differential equations. This book illustrates the basic mathematical concepts and looks at a large variety of boundary value problems.

The Mathematical Foundation of Domain Decomposition Methods

(40)

Multi-domain formulation and the Steklov--Poincare interface equation

(3)

Variational formulation of the multi-domain problem

(5)

Iterative substructuring methods based on transmission conditions at the interface

(8)

Generalisations

(8)

The Steklov--Poincare equation for the Neumann boundary value problem

(2)

Iterations on many subdomains

(2)

The Schwarz method for overlapping subdomains

(8)

The multiplicative and additive forms of the Schwarz method

(2)

Variational interpretation of the Schwarz method

(1)

The Schwarz method as a projection method

(2)

The Schwarz method as a Richardson method

(1)

A characterisation of the projection operators

(1)

The Schwarz method for many subdomains

(1)

The fictitious domain method

(4)

The three-field method

(3)

Discretised Equations and Domain Decomposition Methods

(30)

Finite element approximation of elliptic equations

(5)

The multi-domain formulation for finite elements

(2)

Algebraic formulation of the discrete problem

(1)

Finite element approximation of the Steklov--Poincare operator

(3)

Eigenvalue analysis for the finite element Steklov--Poincare operator

(1)

Algebraic formulation of the discrete Steklov--Poincare operator: the Schur complement matrix

(6)

Preconditioners of the stiffness matrix derived from preconditioners of the Schur complement matrix

(4)

The case of many subdomains

(4)

Non-conforming domain decomposition methods

(12)

The mortar method

(6)

The three-field method at the finite dimensional level

(5)

Iterative Domain Decomposition Methods at the Discrete Level

(32)

Iterative substructuring methods at the finite element level

(2)

The link between the Schur complement system and iterative substructuring methods

(4)

Schur complement preconditioners

(9)

Decomposition with two subdomains

(2)

Decomposition with many subdomains

(7)

The Schwarz method for finite elements

(5)

Acceleration of the Schwarz method

(5)

Inexact solvers

(1)

Two-level methods

(4)

Abstract setting of two-level methods

(1)

Multiplicative and additive two-level preconditioners

(1)

The case of the Schwarz method

(1)

Convergence of two-level methods

(1)

Direct Galerkin approximation of the Steklov--Poincare equation

100

(3)

Convergence Analysis for Iterative Domain Decomposition Algorithms

103

(38)

Extension theorems and spectrally equivalent operators

104

(13)

Extension theorems in H1 (Ωi)

104

(7)

Extension theorems in H (div; Ωi)

111

(3)

Extension theorems in H (rot; Ωi)

114

(3)

Splitting of operators and preconditioned iterative methods

117

(16)

The finite dimensional case

122

(3)

The case of symmetric matrices

125

(3)

The case of complex matrices

128

(5)

Convergence of the Dirichlet--Neumann iterative method

133

(2)

An alternative way to prove convergence

133

(2)

Convergence of the Neumann--Neumann iterative method

135

(1)

Convergence of the Robin iterative method

135

(2)

Convergence of the alternating Schwarz method

137

(4)

Other Boundary Value Problems

141

(78)

Non-symmetric elliptic operators

141

(6)

Weak multi-domain formulation and the Steklov--Poincare interface equation

142

(4)

Substructuring iterative methods

146

(1)

The finite dimensional approximation

147

(1)

The problem of linear elasticity

147

(6)

Weak multi-domain formulation and the Steklov--Poincare interface equation

148

(3)

Substructuring iterative methods

151

(1)

The finite dimensional approximation

151

(2)

The Stokes problem

153

(37)

Weak multi-domain formulation and the Steklov--Poincare interface equation

156

(14)

Substructuring iterative methods

170

(3)

Finite dimensional approximation: the case of discontinuous pressure

173

(12)

Finite dimensional approximation: the case of continuous pressure

185

(4)

Methods based on the Uzawa pressure operator

189

(1)

The Stokes problem for compressible flows

190

(7)

Weak multi-domain formulation and the Steklov--Poincare interface equation

191

(3)

Substructuring iterative methods

194

(1)

The finite dimensional approximation

195

(2)

The Stokes problem for inviscid compressible flows

197

(6)

Weak multi-domain formulation and the Steklov--Poincare interface equation

198

(3)

Substructuring iterative methods

201

(1)

The finite dimensional approximation

202

(1)

First-order equations

203

(7)

Weak multi-domain formulation and the Steklov--Poincare interface equation

204

(4)

Substructuring iterative methods

208

(2)

The time-harmonic Maxwell equations

210

(9)

Weak multi-domain formulation

211

(2)

The finite dimensional Steklov--Poincare interface equation

213

(1)

Substructuring iterative methods

214

(5)

Advection--Diffusion Equations

219

(32)

The advection--diffusion problem and its multi-domain formulations

220

(4)

Iterative substructuring methods for one-dimensional problems

224

(3)

Adaptive iterative substructuring methods: ADN, ARN and ARβN

227

(7)

The damped form of the iterative algorithms: d-ADN, d-ARN and d-ARβN

232

(2)

Coercive iterative substructuring methods: γ-DR and γ-RR

234

(10)

The γ-DR iterative method

234

(3)

Convergence of the γ-DR iterative method

237

(2)

The γ-DR iterative method for systems of advection--diffusion equations

239

(1)

The γ-RR iterative method

240

(2)

Convergence of the γ-RR iterative method

242

(2)

The finite element realisation of the iterative algorithms

244

(7)

Time-Dependent Problems

251

(34)

Parabolic problems

252

(9)

Multi-domain formulation and space discretisation

253

(3)

Implicit time discretisation and subdomain iterations

256

(5)

Hyperbolic problems

261

(11)

Multi-domain formulation

263

(3)

Implicit time discretisation and subdomain iterations

266

(6)

Non-linear time-dependent problems

272

(13)

Navier--Stokes equations for incompressible flows

273

(2)

Navier--Stokes equations for compressible flows

275

(2)

Euler equations for compressible flows

277

(8)

Heterogeneous Domain Decomposition Methods

285

(48)

Heterogeneous models for advection-diffusion equations

287

(9)

The Steklov--Poincare reformulation

290

(5)

The coupling for non-linear convection-diffusion equations

295

(1)

Heterogeneous models for incompressible flows

296

(9)

The coupling for the linearised Stokes problem

297

(3)

The coupling for the Navier--Stokes equations in exterior domains

300

(5)

Heterogeneous models for compressible flows

305

(12)

The coupling between the Navier--Stokes and Euler equations

306

(3)

The coupling between the Euler equations and the full potential equation

309

(8)

The coupling for the compressible Stokes equations

317

(11)

Variational formulation and finite element approximation

320

(2)

An alternative formulation: variational setting and finite element approximation

322

(6)

The coupling for the time-harmonic Maxwell equations

328

(5)

Appendix

333

(10)

Function spaces

333

(6)

Some properties of the Sobolev spaces

339

(4)

References

343

(14)

Index

357

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Domain Decomposition Methods for Partial Differential Equations

Summary

Table of Contents

Supplemental Materials

FREE SHIPPING

Rent More, Save More! Use code: BBRENTAL

5% off 1 book, 7% off 2 books, 10% off 3+ books

Domain Decomposition Methods for Partial Differential Equations

Summary

Table of Contents

Supplemental Materials

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