Domain Decomposition Methods for Partial Differential Equations

by Quarteroni, Alfio; Valli, Alberto

• ISBN13: 9780198501787
• ISBN10: 0198501781
• Format: Hardcover
• Copyright: 1999-07-29
• Publisher: Oxford University Press

Summary

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.

Table of Contents

The Mathematical Foundation of Domain Decomposition Methods	1	(40)
Multi-domain formulation and the Steklov--Poincare interface equation	2	(3)
Variational formulation of the multi-domain problem	5	(5)
Iterative substructuring methods based on transmission conditions at the interface	10	(8)
Generalisations	18	(8)
The Steklov--Poincare equation for the Neumann boundary value problem	22	(2)
Iterations on many subdomains	24	(2)
The Schwarz method for overlapping subdomains	26	(8)
The multiplicative and additive forms of the Schwarz method	26	(2)
Variational interpretation of the Schwarz method	28	(1)
The Schwarz method as a projection method	29	(2)
The Schwarz method as a Richardson method	31	(1)
A characterisation of the projection operators	32	(1)
The Schwarz method for many subdomains	33	(1)
The fictitious domain method	34	(4)
The three-field method	38	(3)
Discretised Equations and Domain Decomposition Methods	41	(30)
Finite element approximation of elliptic equations	41	(5)
The multi-domain formulation for finite elements	43	(2)
Algebraic formulation of the discrete problem	45	(1)
Finite element approximation of the Steklov--Poincare operator	46	(3)
Eigenvalue analysis for the finite element Steklov--Poincare operator	48	(1)
Algebraic formulation of the discrete Steklov--Poincare operator: the Schur complement matrix	49	(6)
Preconditioners of the stiffness matrix derived from preconditioners of the Schur complement matrix	51	(4)
The case of many subdomains	55	(4)
Non-conforming domain decomposition methods	59	(12)
The mortar method	60	(6)
The three-field method at the finite dimensional level	66	(5)
Iterative Domain Decomposition Methods at the Discrete Level	71	(32)
Iterative substructuring methods at the finite element level	71	(2)
The link between the Schur complement system and iterative substructuring methods	73	(4)
Schur complement preconditioners	77	(9)
Decomposition with two subdomains	77	(2)
Decomposition with many subdomains	79	(7)
The Schwarz method for finite elements	86	(5)
Acceleration of the Schwarz method	91	(5)
Inexact solvers	96	(1)
Two-level methods	96	(4)
Abstract setting of two-level methods	97	(1)
Multiplicative and additive two-level preconditioners	98	(1)
The case of the Schwarz method	99	(1)
Convergence of two-level methods	99	(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