Domain Decomposition Methods. Partial Differential Equations


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1 Domain Decomposition Methods for Partial Differential Equations ALFIO QUARTERONI Professor ofnumericalanalysis, Politecnico di Milano, Italy, and Ecole Polytechnique Federale de Lausanne, Switzerland ALBERTO VALLI Professor offunctionalanalysis, University oftrento, Italy CLARENDON PRESS OXFORD 1999
2 CONTENTS 1 The Mathematical Foundation of Domain Decomposition Methods Multidomain formulation and the SteklovPoincare interface equation Variational formulation of the multidomain problem Iterative substructuring methods based on transmission conditions at the interface Generalisations The SteklovPoincare equation for the Neumann boundary value problem Iterations on many subdomains The Schwarz method for overlapping subdomains The multiplicative and additive forms of the Schwarz method Variational Interpretation of the Schwarz method The Schwarz method as a projection method The Schwarz method as a Richardson method A characterisation of the projection Operators The Schwarz method for many subdomains The fictitious domain method The threefield method 38 2 Discretised Equations and Domain Decomposition Methods Finite element approximation of elliptic equations The multidomain formulation for finite elements Algebraic formulation of the discrete problem Finite element approximation of the SteklovPoincare Operator Eigenvalue analysis for the finite element Steklov Poincare Operator Algebraic formulation of the discrete SteklovPoincare Operator: the Schur complement matrix Preconditioners of the stiffness matrix derived from preconditioners of the Schur complement matrix The case of many subdomains 55
3 Xll CONTENTS 2.5 Nonconforming domain decomposition methods The mortar method The threefield method at the finite dimensional level 66 3 Iterative Domain Decomposition Methods at the Discrete Level Iterative substructuring methods at the finite element level The link between the Schur complement System and iterative substructuring methods Schur complement preconditioners Decomposition with two subdomains Decomposition with many subdomains The Schwarz method for finite elements Acceleration of the Schwarz method Inexact solvers Twolevel methods Abstract setting of twolevel methods Multiplicative and additive twolevel preconditioners The case of the Schwarz method Convergence of twolevel methods Direct Galerkin approximation of the SteklovPoincare equation Convergence Analysis for Iterative Domain Decomposition Algorithms Extension theorems and spectrally equivalent Operators Extension theorems in ij 1 (ft i ) Extension theorems in ij(div; fij) Extension theorems in H(rot; fl^) Splitting of Operators and preconditioned iterative methods The finite dimensional case The case of Symmetrie matrices The case of complex matrices Convergence of the DirichletNeumann iterative method An alternative way to prove convergence Convergence of the NeumannNeumann iterative method Convergence of the Robin iterative method Convergence of the alternating Schwarz method Other Boundary Value Problems Nonsymmetric elliptic Operators 141
4 CONTENTS xm Weak multidomain formulation and the Steklov Poincare interface equation Substructuring iterative methods The finite dimensional approximation The problem of linear elasticity Weak multidomain formulation and the Steklov Poincare interface equation Substructuring iterative methods The finite dimensional approximation The Stokes problem Weak multidomain formulation and the Steklov Poincare interface equation Substructuring iterative methods Finite dimensional approximation: the case of discontinuous pressure Finite dimensional approximation: the case of continuous pressure Methods based on the Uzawa pressure Operator The Stokes problem for compressible flows Weak multidomain formulation and the Steklov Poincare interface equation Substructuring iterative methods The finite dimensional approximation The Stokes problem for inviscid compressible flows Weak multidomain formulation and the Steklov Poincare interface equation Substructuring iterative methods The finite dimensional approximation Firstorder equations Weak multidomain formulation and the Steklov Poincare interface equation Substructuring iterative methods The timeharmonic Maxwell equations Weak multidomain formulation The finite dimensional SteklovPoincare interface equation Substructuring iterative methods Advection Diffusion Equations The advectiondiffusion problem and its multidomain formulations Iterative substructuring methods for onedimensional Problems Adaptive iterative substructuring methods: ADN, ARN and AR^N 227
5 XIV CONTENTS The damped form of the iterative algorithms: d ADN, darn and dar^n Coercive iterative substructuring methods: 7DR and 7 RR The 7DR iterative method Convergence of the 7DR iterative method The 7DR iterative method for Systems of advectiondiffusion equations The 7RR iterative method Convergence of the 7RR iterative method The finite element realisation of the iterative algorithms TimeDependent Problems Parabolic problems Multidomain formulation and space discretisation Implicit time discretisation and subdomain iterations Hyperbolic problems Multidomain formulation Implicit time discretisation and subdomain iterations Nonlinear timedependent problems NavierStokes equations for incompressible flows NavierStokes equations for compressible flows Euler equations for compressible flows Heterogeneous Domain Decomposition Methods Heterogeneous modeis for advectiondiffusion equations The SteklovPoincare reformulation The coupling for nonlinear convectiondiffusion equations Heterogeneous modeis for incompressible flows The coupling for the linearised Stokes problem The coupling for the NavierStokes equations in exterior domains Heterogeneous modeis for compressible flows The coupling between the NavierStokes and Euler equations The coupling between the Euler equations and the füll potential equation The coupling for the compressible Stokes equations Variational formulation and finite element approximation An alternative formulation: variational setting and finite element approximation 322
6 CONTENTS 8.5 The coupling for the timeharmonic Maxwell equations Appendix Function Spaces Some properties of the Sobolev Spaces 339 References 343 Index 357
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